Copied to
clipboard

## G = D7×C25order 448 = 26·7

### Direct product of C25 and D7

Aliases: D7×C25, C7⋊C26, C14⋊C25, (C2×C14)⋊4C24, (C24×C14)⋊5C2, (C23×C14)⋊20C22, (C22×C14)⋊10C23, SmallGroup(448,1395)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C25
 Chief series C1 — C7 — D7 — D14 — C22×D7 — C23×D7 — D7×C24 — D7×C25
 Lower central C7 — D7×C25
 Upper central C1 — C25

Generators and relations for D7×C25
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f7=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 20356 in 5650 conjugacy classes, 3199 normal (5 characteristic)
C1, C2, C2, C22, C22, C7, C23, C23, D7, C14, C24, C24, D14, C2×C14, C25, C25, C22×D7, C22×C14, C26, C23×D7, C23×C14, D7×C24, C24×C14, D7×C25
Quotients: C1, C2, C22, C23, D7, C24, D14, C25, C22×D7, C26, C23×D7, D7×C24, D7×C25

Smallest permutation representation of D7×C25
On 224 points
Generators in S224
(1 223)(2 224)(3 218)(4 219)(5 220)(6 221)(7 222)(8 211)(9 212)(10 213)(11 214)(12 215)(13 216)(14 217)(15 204)(16 205)(17 206)(18 207)(19 208)(20 209)(21 210)(22 197)(23 198)(24 199)(25 200)(26 201)(27 202)(28 203)(29 190)(30 191)(31 192)(32 193)(33 194)(34 195)(35 196)(36 183)(37 184)(38 185)(39 186)(40 187)(41 188)(42 189)(43 176)(44 177)(45 178)(46 179)(47 180)(48 181)(49 182)(50 169)(51 170)(52 171)(53 172)(54 173)(55 174)(56 175)(57 162)(58 163)(59 164)(60 165)(61 166)(62 167)(63 168)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 161)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 141)(79 142)(80 143)(81 144)(82 145)(83 146)(84 147)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 127)(93 128)(94 129)(95 130)(96 131)(97 132)(98 133)(99 120)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 113)(107 114)(108 115)(109 116)(110 117)(111 118)(112 119)
(1 83)(2 84)(3 78)(4 79)(5 80)(6 81)(7 82)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 99)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(113 190)(114 191)(115 192)(116 193)(117 194)(118 195)(119 196)(120 183)(121 184)(122 185)(123 186)(124 187)(125 188)(126 189)(127 176)(128 177)(129 178)(130 179)(131 180)(132 181)(133 182)(134 169)(135 170)(136 171)(137 172)(138 173)(139 174)(140 175)(141 218)(142 219)(143 220)(144 221)(145 222)(146 223)(147 224)(148 211)(149 212)(150 213)(151 214)(152 215)(153 216)(154 217)(155 204)(156 205)(157 206)(158 207)(159 208)(160 209)(161 210)(162 197)(163 198)(164 199)(165 200)(166 201)(167 202)(168 203)
(1 41)(2 42)(3 36)(4 37)(5 38)(6 39)(7 40)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(113 148)(114 149)(115 150)(116 151)(117 152)(118 153)(119 154)(120 141)(121 142)(122 143)(123 144)(124 145)(125 146)(126 147)(127 162)(128 163)(129 164)(130 165)(131 166)(132 167)(133 168)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 161)(169 204)(170 205)(171 206)(172 207)(173 208)(174 209)(175 210)(176 197)(177 198)(178 199)(179 200)(180 201)(181 202)(182 203)(183 218)(184 219)(185 220)(186 221)(187 222)(188 223)(189 224)(190 211)(191 212)(192 213)(193 214)(194 215)(195 216)(196 217)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)
(1 166)(2 165)(3 164)(4 163)(5 162)(6 168)(7 167)(8 157)(9 156)(10 155)(11 161)(12 160)(13 159)(14 158)(15 150)(16 149)(17 148)(18 154)(19 153)(20 152)(21 151)(22 143)(23 142)(24 141)(25 147)(26 146)(27 145)(28 144)(29 136)(30 135)(31 134)(32 140)(33 139)(34 138)(35 137)(36 129)(37 128)(38 127)(39 133)(40 132)(41 131)(42 130)(43 122)(44 121)(45 120)(46 126)(47 125)(48 124)(49 123)(50 115)(51 114)(52 113)(53 119)(54 118)(55 117)(56 116)(57 220)(58 219)(59 218)(60 224)(61 223)(62 222)(63 221)(64 213)(65 212)(66 211)(67 217)(68 216)(69 215)(70 214)(71 206)(72 205)(73 204)(74 210)(75 209)(76 208)(77 207)(78 199)(79 198)(80 197)(81 203)(82 202)(83 201)(84 200)(85 192)(86 191)(87 190)(88 196)(89 195)(90 194)(91 193)(92 185)(93 184)(94 183)(95 189)(96 188)(97 187)(98 186)(99 178)(100 177)(101 176)(102 182)(103 181)(104 180)(105 179)(106 171)(107 170)(108 169)(109 175)(110 174)(111 173)(112 172)

G:=sub<Sym(224)| (1,223)(2,224)(3,218)(4,219)(5,220)(6,221)(7,222)(8,211)(9,212)(10,213)(11,214)(12,215)(13,216)(14,217)(15,204)(16,205)(17,206)(18,207)(19,208)(20,209)(21,210)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,190)(30,191)(31,192)(32,193)(33,194)(34,195)(35,196)(36,183)(37,184)(38,185)(39,186)(40,187)(41,188)(42,189)(43,176)(44,177)(45,178)(46,179)(47,180)(48,181)(49,182)(50,169)(51,170)(52,171)(53,172)(54,173)(55,174)(56,175)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,161)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,113)(107,114)(108,115)(109,116)(110,117)(111,118)(112,119), (1,83)(2,84)(3,78)(4,79)(5,80)(6,81)(7,82)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(113,190)(114,191)(115,192)(116,193)(117,194)(118,195)(119,196)(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,176)(128,177)(129,178)(130,179)(131,180)(132,181)(133,182)(134,169)(135,170)(136,171)(137,172)(138,173)(139,174)(140,175)(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,217)(155,204)(156,205)(157,206)(158,207)(159,208)(160,209)(161,210)(162,197)(163,198)(164,199)(165,200)(166,201)(167,202)(168,203), (1,41)(2,42)(3,36)(4,37)(5,38)(6,39)(7,40)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(169,204)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,197)(177,198)(178,199)(179,200)(180,201)(181,202)(182,203)(183,218)(184,219)(185,220)(186,221)(187,222)(188,223)(189,224)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,166)(2,165)(3,164)(4,163)(5,162)(6,168)(7,167)(8,157)(9,156)(10,155)(11,161)(12,160)(13,159)(14,158)(15,150)(16,149)(17,148)(18,154)(19,153)(20,152)(21,151)(22,143)(23,142)(24,141)(25,147)(26,146)(27,145)(28,144)(29,136)(30,135)(31,134)(32,140)(33,139)(34,138)(35,137)(36,129)(37,128)(38,127)(39,133)(40,132)(41,131)(42,130)(43,122)(44,121)(45,120)(46,126)(47,125)(48,124)(49,123)(50,115)(51,114)(52,113)(53,119)(54,118)(55,117)(56,116)(57,220)(58,219)(59,218)(60,224)(61,223)(62,222)(63,221)(64,213)(65,212)(66,211)(67,217)(68,216)(69,215)(70,214)(71,206)(72,205)(73,204)(74,210)(75,209)(76,208)(77,207)(78,199)(79,198)(80,197)(81,203)(82,202)(83,201)(84,200)(85,192)(86,191)(87,190)(88,196)(89,195)(90,194)(91,193)(92,185)(93,184)(94,183)(95,189)(96,188)(97,187)(98,186)(99,178)(100,177)(101,176)(102,182)(103,181)(104,180)(105,179)(106,171)(107,170)(108,169)(109,175)(110,174)(111,173)(112,172)>;

G:=Group( (1,223)(2,224)(3,218)(4,219)(5,220)(6,221)(7,222)(8,211)(9,212)(10,213)(11,214)(12,215)(13,216)(14,217)(15,204)(16,205)(17,206)(18,207)(19,208)(20,209)(21,210)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,190)(30,191)(31,192)(32,193)(33,194)(34,195)(35,196)(36,183)(37,184)(38,185)(39,186)(40,187)(41,188)(42,189)(43,176)(44,177)(45,178)(46,179)(47,180)(48,181)(49,182)(50,169)(51,170)(52,171)(53,172)(54,173)(55,174)(56,175)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,161)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,113)(107,114)(108,115)(109,116)(110,117)(111,118)(112,119), (1,83)(2,84)(3,78)(4,79)(5,80)(6,81)(7,82)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,106)(30,107)(31,108)(32,109)(33,110)(34,111)(35,112)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(113,190)(114,191)(115,192)(116,193)(117,194)(118,195)(119,196)(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,176)(128,177)(129,178)(130,179)(131,180)(132,181)(133,182)(134,169)(135,170)(136,171)(137,172)(138,173)(139,174)(140,175)(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,217)(155,204)(156,205)(157,206)(158,207)(159,208)(160,209)(161,210)(162,197)(163,198)(164,199)(165,200)(166,201)(167,202)(168,203), (1,41)(2,42)(3,36)(4,37)(5,38)(6,39)(7,40)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(169,204)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,197)(177,198)(178,199)(179,200)(180,201)(181,202)(182,203)(183,218)(184,219)(185,220)(186,221)(187,222)(188,223)(189,224)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,166)(2,165)(3,164)(4,163)(5,162)(6,168)(7,167)(8,157)(9,156)(10,155)(11,161)(12,160)(13,159)(14,158)(15,150)(16,149)(17,148)(18,154)(19,153)(20,152)(21,151)(22,143)(23,142)(24,141)(25,147)(26,146)(27,145)(28,144)(29,136)(30,135)(31,134)(32,140)(33,139)(34,138)(35,137)(36,129)(37,128)(38,127)(39,133)(40,132)(41,131)(42,130)(43,122)(44,121)(45,120)(46,126)(47,125)(48,124)(49,123)(50,115)(51,114)(52,113)(53,119)(54,118)(55,117)(56,116)(57,220)(58,219)(59,218)(60,224)(61,223)(62,222)(63,221)(64,213)(65,212)(66,211)(67,217)(68,216)(69,215)(70,214)(71,206)(72,205)(73,204)(74,210)(75,209)(76,208)(77,207)(78,199)(79,198)(80,197)(81,203)(82,202)(83,201)(84,200)(85,192)(86,191)(87,190)(88,196)(89,195)(90,194)(91,193)(92,185)(93,184)(94,183)(95,189)(96,188)(97,187)(98,186)(99,178)(100,177)(101,176)(102,182)(103,181)(104,180)(105,179)(106,171)(107,170)(108,169)(109,175)(110,174)(111,173)(112,172) );

G=PermutationGroup([[(1,223),(2,224),(3,218),(4,219),(5,220),(6,221),(7,222),(8,211),(9,212),(10,213),(11,214),(12,215),(13,216),(14,217),(15,204),(16,205),(17,206),(18,207),(19,208),(20,209),(21,210),(22,197),(23,198),(24,199),(25,200),(26,201),(27,202),(28,203),(29,190),(30,191),(31,192),(32,193),(33,194),(34,195),(35,196),(36,183),(37,184),(38,185),(39,186),(40,187),(41,188),(42,189),(43,176),(44,177),(45,178),(46,179),(47,180),(48,181),(49,182),(50,169),(51,170),(52,171),(53,172),(54,173),(55,174),(56,175),(57,162),(58,163),(59,164),(60,165),(61,166),(62,167),(63,168),(64,155),(65,156),(66,157),(67,158),(68,159),(69,160),(70,161),(71,148),(72,149),(73,150),(74,151),(75,152),(76,153),(77,154),(78,141),(79,142),(80,143),(81,144),(82,145),(83,146),(84,147),(85,134),(86,135),(87,136),(88,137),(89,138),(90,139),(91,140),(92,127),(93,128),(94,129),(95,130),(96,131),(97,132),(98,133),(99,120),(100,121),(101,122),(102,123),(103,124),(104,125),(105,126),(106,113),(107,114),(108,115),(109,116),(110,117),(111,118),(112,119)], [(1,83),(2,84),(3,78),(4,79),(5,80),(6,81),(7,82),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,106),(30,107),(31,108),(32,109),(33,110),(34,111),(35,112),(36,99),(37,100),(38,101),(39,102),(40,103),(41,104),(42,105),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91),(113,190),(114,191),(115,192),(116,193),(117,194),(118,195),(119,196),(120,183),(121,184),(122,185),(123,186),(124,187),(125,188),(126,189),(127,176),(128,177),(129,178),(130,179),(131,180),(132,181),(133,182),(134,169),(135,170),(136,171),(137,172),(138,173),(139,174),(140,175),(141,218),(142,219),(143,220),(144,221),(145,222),(146,223),(147,224),(148,211),(149,212),(150,213),(151,214),(152,215),(153,216),(154,217),(155,204),(156,205),(157,206),(158,207),(159,208),(160,209),(161,210),(162,197),(163,198),(164,199),(165,200),(166,201),(167,202),(168,203)], [(1,41),(2,42),(3,36),(4,37),(5,38),(6,39),(7,40),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(57,92),(58,93),(59,94),(60,95),(61,96),(62,97),(63,98),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(113,148),(114,149),(115,150),(116,151),(117,152),(118,153),(119,154),(120,141),(121,142),(122,143),(123,144),(124,145),(125,146),(126,147),(127,162),(128,163),(129,164),(130,165),(131,166),(132,167),(133,168),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,161),(169,204),(170,205),(171,206),(172,207),(173,208),(174,209),(175,210),(176,197),(177,198),(178,199),(179,200),(180,201),(181,202),(182,203),(183,218),(184,219),(185,220),(186,221),(187,222),(188,223),(189,224),(190,211),(191,212),(192,213),(193,214),(194,215),(195,216),(196,217)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(169,183),(170,184),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139),(133,140),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167),(161,168),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210),(211,218),(212,219),(213,220),(214,221),(215,222),(216,223),(217,224)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,166),(2,165),(3,164),(4,163),(5,162),(6,168),(7,167),(8,157),(9,156),(10,155),(11,161),(12,160),(13,159),(14,158),(15,150),(16,149),(17,148),(18,154),(19,153),(20,152),(21,151),(22,143),(23,142),(24,141),(25,147),(26,146),(27,145),(28,144),(29,136),(30,135),(31,134),(32,140),(33,139),(34,138),(35,137),(36,129),(37,128),(38,127),(39,133),(40,132),(41,131),(42,130),(43,122),(44,121),(45,120),(46,126),(47,125),(48,124),(49,123),(50,115),(51,114),(52,113),(53,119),(54,118),(55,117),(56,116),(57,220),(58,219),(59,218),(60,224),(61,223),(62,222),(63,221),(64,213),(65,212),(66,211),(67,217),(68,216),(69,215),(70,214),(71,206),(72,205),(73,204),(74,210),(75,209),(76,208),(77,207),(78,199),(79,198),(80,197),(81,203),(82,202),(83,201),(84,200),(85,192),(86,191),(87,190),(88,196),(89,195),(90,194),(91,193),(92,185),(93,184),(94,183),(95,189),(96,188),(97,187),(98,186),(99,178),(100,177),(101,176),(102,182),(103,181),(104,180),(105,179),(106,171),(107,170),(108,169),(109,175),(110,174),(111,173),(112,172)]])

160 conjugacy classes

 class 1 2A ··· 2AE 2AF ··· 2BK 7A 7B 7C 14A ··· 14CO order 1 2 ··· 2 2 ··· 2 7 7 7 14 ··· 14 size 1 1 ··· 1 7 ··· 7 2 2 2 2 ··· 2

160 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D7 D14 kernel D7×C25 D7×C24 C24×C14 C25 C24 # reps 1 62 1 3 93

Matrix representation of D7×C25 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 28 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 1 0 0 0 0 28 0
,
 28 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 25 4 0 0 0 0 18 4

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,28,0,0,0,0,1,0],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,25,18,0,0,0,0,4,4] >;

D7×C25 in GAP, Magma, Sage, TeX

D_7\times C_2^5
% in TeX

G:=Group("D7xC2^5");
// GroupNames label

G:=SmallGroup(448,1395);
// by ID

G=gap.SmallGroup(448,1395);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^7=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,g*f*g=f^-1>;
// generators/relations

׿
×
𝔽