direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.25D4, (C2×C8)⋊6C28, (C2×C56)⋊16C4, C4.3(Q8×C14), C56.84(C2×C4), C8.16(C2×C28), C2.D8⋊13C14, C4.Q8⋊13C14, C28.70(C4⋊C4), (C2×C28).78Q8, C28.92(C2×Q8), (C2×C28).537D4, C23.24(C7×D4), (C22×C8).11C14, C4.26(C22×C28), (C22×C56).29C2, C22.49(D4×C14), C14.115(C4○D8), C42⋊C2.6C14, (C2×C28).900C23, C28.184(C22×C4), (C2×C56).436C22, (C22×C14).128D4, (C22×C28).589C22, C4.21(C7×C4⋊C4), C2.2(C7×C4○D8), C2.13(C14×C4⋊C4), C14.69(C2×C4⋊C4), (C7×C2.D8)⋊28C2, (C7×C4.Q8)⋊28C2, C22.9(C7×C4⋊C4), (C2×C4).20(C7×Q8), C4⋊C4.43(C2×C14), (C2×C4).76(C2×C28), (C2×C8).76(C2×C14), (C2×C4).147(C7×D4), (C2×C14).26(C4⋊C4), (C2×C28).337(C2×C4), (C2×C14).625(C2×D4), (C7×C4⋊C4).364C22, (C2×C4).75(C22×C14), (C22×C4).118(C2×C14), (C7×C42⋊C2).20C2, SmallGroup(448,835)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.25D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=1, e4=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >
Subgroups: 162 in 114 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C4.Q8, C2.D8, C42⋊C2, C22×C8, C56, C2×C28, C2×C28, C2×C28, C22×C14, C23.25D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C22×C28, C7×C4.Q8, C7×C2.D8, C7×C42⋊C2, C22×C56, C7×C23.25D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, C22×C4, C2×D4, C2×Q8, C28, C2×C14, C2×C4⋊C4, C4○D8, C2×C28, C7×D4, C7×Q8, C22×C14, C23.25D4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C14×C4⋊C4, C7×C4○D8, C7×C23.25D4
►On 224 points
Generators in S224
(1 14 71 87 31 79 23)(2 15 72 88 32 80 24)(3 16 65 81 25 73 17)(4 9 66 82 26 74 18)(5 10 67 83 27 75 19)(6 11 68 84 28 76 20)(7 12 69 85 29 77 21)(8 13 70 86 30 78 22)(33 117 105 49 97 41 89)(34 118 106 50 98 42 90)(35 119 107 51 99 43 91)(36 120 108 52 100 44 92)(37 113 109 53 101 45 93)(38 114 110 54 102 46 94)(39 115 111 55 103 47 95)(40 116 112 56 104 48 96)(57 219 167 211 159 203 151)(58 220 168 212 160 204 152)(59 221 161 213 153 205 145)(60 222 162 214 154 206 146)(61 223 163 215 155 207 147)(62 224 164 216 156 208 148)(63 217 165 209 157 201 149)(64 218 166 210 158 202 150)(121 173 200 140 192 132 184)(122 174 193 141 185 133 177)(123 175 194 142 186 134 178)(124 176 195 143 187 135 179)(125 169 196 144 188 136 180)(126 170 197 137 189 129 181)(127 171 198 138 190 130 182)(128 172 199 139 191 131 183)
(57 61)(58 62)(59 63)(60 64)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)(161 165)(162 166)(163 167)(164 168)(169 173)(170 174)(171 175)(172 176)(177 181)(178 182)(179 183)(180 184)(185 189)(186 190)(187 191)(188 192)(193 197)(194 198)(195 199)(196 200)(201 205)(202 206)(203 207)(204 208)(209 213)(210 214)(211 215)(212 216)(217 221)(218 222)(219 223)(220 224)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 118)(10 119)(11 120)(12 113)(13 114)(14 115)(15 116)(16 117)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 128)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(129 201)(130 202)(131 203)(132 204)(133 205)(134 206)(135 207)(136 208)(137 209)(138 210)(139 211)(140 212)(141 213)(142 214)(143 215)(144 216)(145 177)(146 178)(147 179)(148 180)(149 181)(150 182)(151 183)(152 184)(153 185)(154 186)(155 187)(156 188)(157 189)(158 190)(159 191)(160 192)(161 193)(162 194)(163 195)(164 196)(165 197)(166 198)(167 199)(168 200)(169 224)(170 217)(171 218)(172 219)(173 220)(174 221)(175 222)(176 223)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)(161 165)(162 166)(163 167)(164 168)(169 173)(170 174)(171 175)(172 176)(177 181)(178 182)(179 183)(180 184)(185 189)(186 190)(187 191)(188 192)(193 197)(194 198)(195 199)(196 200)(201 205)(202 206)(203 207)(204 208)(209 213)(210 214)(211 215)(212 216)(217 221)(218 222)(219 223)(220 224)
(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)
(1 148 35 184)(2 151 36 179)(3 146 37 182)(4 149 38 177)(5 152 39 180)(6 147 40 183)(7 150 33 178)(8 145 34 181)(9 63 114 122)(10 58 115 125)(11 61 116 128)(12 64 117 123)(13 59 118 126)(14 62 119 121)(15 57 120 124)(16 60 113 127)(17 206 93 130)(18 201 94 133)(19 204 95 136)(20 207 96 131)(21 202 89 134)(22 205 90 129)(23 208 91 132)(24 203 92 135)(25 214 101 138)(26 209 102 141)(27 212 103 144)(28 215 104 139)(29 210 97 142)(30 213 98 137)(31 216 99 140)(32 211 100 143)(41 186 77 158)(42 189 78 153)(43 192 79 156)(44 187 80 159)(45 190 73 154)(46 185 74 157)(47 188 75 160)(48 191 76 155)(49 194 85 166)(50 197 86 161)(51 200 87 164)(52 195 88 167)(53 198 81 162)(54 193 82 165)(55 196 83 168)(56 199 84 163)(65 222 109 171)(66 217 110 174)(67 220 111 169)(68 223 112 172)(69 218 105 175)(70 221 106 170)(71 224 107 173)(72 219 108 176)
G:=sub<Sym(224)| (1,14,71,87,31,79,23)(2,15,72,88,32,80,24)(3,16,65,81,25,73,17)(4,9,66,82,26,74,18)(5,10,67,83,27,75,19)(6,11,68,84,28,76,20)(7,12,69,85,29,77,21)(8,13,70,86,30,78,22)(33,117,105,49,97,41,89)(34,118,106,50,98,42,90)(35,119,107,51,99,43,91)(36,120,108,52,100,44,92)(37,113,109,53,101,45,93)(38,114,110,54,102,46,94)(39,115,111,55,103,47,95)(40,116,112,56,104,48,96)(57,219,167,211,159,203,151)(58,220,168,212,160,204,152)(59,221,161,213,153,205,145)(60,222,162,214,154,206,146)(61,223,163,215,155,207,147)(62,224,164,216,156,208,148)(63,217,165,209,157,201,149)(64,218,166,210,158,202,150)(121,173,200,140,192,132,184)(122,174,193,141,185,133,177)(123,175,194,142,186,134,178)(124,176,195,143,187,135,179)(125,169,196,144,188,136,180)(126,170,197,137,189,129,181)(127,171,198,138,190,130,182)(128,172,199,139,191,131,183), (57,61)(58,62)(59,63)(60,64)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,118)(10,119)(11,120)(12,113)(13,114)(14,115)(15,116)(16,117)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,128)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(129,201)(130,202)(131,203)(132,204)(133,205)(134,206)(135,207)(136,208)(137,209)(138,210)(139,211)(140,212)(141,213)(142,214)(143,215)(144,216)(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,183)(152,184)(153,185)(154,186)(155,187)(156,188)(157,189)(158,190)(159,191)(160,192)(161,193)(162,194)(163,195)(164,196)(165,197)(166,198)(167,199)(168,200)(169,224)(170,217)(171,218)(172,219)(173,220)(174,221)(175,222)(176,223), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224), (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), (1,148,35,184)(2,151,36,179)(3,146,37,182)(4,149,38,177)(5,152,39,180)(6,147,40,183)(7,150,33,178)(8,145,34,181)(9,63,114,122)(10,58,115,125)(11,61,116,128)(12,64,117,123)(13,59,118,126)(14,62,119,121)(15,57,120,124)(16,60,113,127)(17,206,93,130)(18,201,94,133)(19,204,95,136)(20,207,96,131)(21,202,89,134)(22,205,90,129)(23,208,91,132)(24,203,92,135)(25,214,101,138)(26,209,102,141)(27,212,103,144)(28,215,104,139)(29,210,97,142)(30,213,98,137)(31,216,99,140)(32,211,100,143)(41,186,77,158)(42,189,78,153)(43,192,79,156)(44,187,80,159)(45,190,73,154)(46,185,74,157)(47,188,75,160)(48,191,76,155)(49,194,85,166)(50,197,86,161)(51,200,87,164)(52,195,88,167)(53,198,81,162)(54,193,82,165)(55,196,83,168)(56,199,84,163)(65,222,109,171)(66,217,110,174)(67,220,111,169)(68,223,112,172)(69,218,105,175)(70,221,106,170)(71,224,107,173)(72,219,108,176)>;
G:=Group( (1,14,71,87,31,79,23)(2,15,72,88,32,80,24)(3,16,65,81,25,73,17)(4,9,66,82,26,74,18)(5,10,67,83,27,75,19)(6,11,68,84,28,76,20)(7,12,69,85,29,77,21)(8,13,70,86,30,78,22)(33,117,105,49,97,41,89)(34,118,106,50,98,42,90)(35,119,107,51,99,43,91)(36,120,108,52,100,44,92)(37,113,109,53,101,45,93)(38,114,110,54,102,46,94)(39,115,111,55,103,47,95)(40,116,112,56,104,48,96)(57,219,167,211,159,203,151)(58,220,168,212,160,204,152)(59,221,161,213,153,205,145)(60,222,162,214,154,206,146)(61,223,163,215,155,207,147)(62,224,164,216,156,208,148)(63,217,165,209,157,201,149)(64,218,166,210,158,202,150)(121,173,200,140,192,132,184)(122,174,193,141,185,133,177)(123,175,194,142,186,134,178)(124,176,195,143,187,135,179)(125,169,196,144,188,136,180)(126,170,197,137,189,129,181)(127,171,198,138,190,130,182)(128,172,199,139,191,131,183), (57,61)(58,62)(59,63)(60,64)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,118)(10,119)(11,120)(12,113)(13,114)(14,115)(15,116)(16,117)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,128)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,105)(66,106)(67,107)(68,108)(69,109)(70,110)(71,111)(72,112)(129,201)(130,202)(131,203)(132,204)(133,205)(134,206)(135,207)(136,208)(137,209)(138,210)(139,211)(140,212)(141,213)(142,214)(143,215)(144,216)(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,183)(152,184)(153,185)(154,186)(155,187)(156,188)(157,189)(158,190)(159,191)(160,192)(161,193)(162,194)(163,195)(164,196)(165,197)(166,198)(167,199)(168,200)(169,224)(170,217)(171,218)(172,219)(173,220)(174,221)(175,222)(176,223), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224), (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), (1,148,35,184)(2,151,36,179)(3,146,37,182)(4,149,38,177)(5,152,39,180)(6,147,40,183)(7,150,33,178)(8,145,34,181)(9,63,114,122)(10,58,115,125)(11,61,116,128)(12,64,117,123)(13,59,118,126)(14,62,119,121)(15,57,120,124)(16,60,113,127)(17,206,93,130)(18,201,94,133)(19,204,95,136)(20,207,96,131)(21,202,89,134)(22,205,90,129)(23,208,91,132)(24,203,92,135)(25,214,101,138)(26,209,102,141)(27,212,103,144)(28,215,104,139)(29,210,97,142)(30,213,98,137)(31,216,99,140)(32,211,100,143)(41,186,77,158)(42,189,78,153)(43,192,79,156)(44,187,80,159)(45,190,73,154)(46,185,74,157)(47,188,75,160)(48,191,76,155)(49,194,85,166)(50,197,86,161)(51,200,87,164)(52,195,88,167)(53,198,81,162)(54,193,82,165)(55,196,83,168)(56,199,84,163)(65,222,109,171)(66,217,110,174)(67,220,111,169)(68,223,112,172)(69,218,105,175)(70,221,106,170)(71,224,107,173)(72,219,108,176) );
G=PermutationGroup([[(1,14,71,87,31,79,23),(2,15,72,88,32,80,24),(3,16,65,81,25,73,17),(4,9,66,82,26,74,18),(5,10,67,83,27,75,19),(6,11,68,84,28,76,20),(7,12,69,85,29,77,21),(8,13,70,86,30,78,22),(33,117,105,49,97,41,89),(34,118,106,50,98,42,90),(35,119,107,51,99,43,91),(36,120,108,52,100,44,92),(37,113,109,53,101,45,93),(38,114,110,54,102,46,94),(39,115,111,55,103,47,95),(40,116,112,56,104,48,96),(57,219,167,211,159,203,151),(58,220,168,212,160,204,152),(59,221,161,213,153,205,145),(60,222,162,214,154,206,146),(61,223,163,215,155,207,147),(62,224,164,216,156,208,148),(63,217,165,209,157,201,149),(64,218,166,210,158,202,150),(121,173,200,140,192,132,184),(122,174,193,141,185,133,177),(123,175,194,142,186,134,178),(124,176,195,143,187,135,179),(125,169,196,144,188,136,180),(126,170,197,137,189,129,181),(127,171,198,138,190,130,182),(128,172,199,139,191,131,183)], [(57,61),(58,62),(59,63),(60,64),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160),(161,165),(162,166),(163,167),(164,168),(169,173),(170,174),(171,175),(172,176),(177,181),(178,182),(179,183),(180,184),(185,189),(186,190),(187,191),(188,192),(193,197),(194,198),(195,199),(196,200),(201,205),(202,206),(203,207),(204,208),(209,213),(210,214),(211,215),(212,216),(217,221),(218,222),(219,223),(220,224)], [(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,118),(10,119),(11,120),(12,113),(13,114),(14,115),(15,116),(16,117),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,128),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,127),(65,105),(66,106),(67,107),(68,108),(69,109),(70,110),(71,111),(72,112),(129,201),(130,202),(131,203),(132,204),(133,205),(134,206),(135,207),(136,208),(137,209),(138,210),(139,211),(140,212),(141,213),(142,214),(143,215),(144,216),(145,177),(146,178),(147,179),(148,180),(149,181),(150,182),(151,183),(152,184),(153,185),(154,186),(155,187),(156,188),(157,189),(158,190),(159,191),(160,192),(161,193),(162,194),(163,195),(164,196),(165,197),(166,198),(167,199),(168,200),(169,224),(170,217),(171,218),(172,219),(173,220),(174,221),(175,222),(176,223)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160),(161,165),(162,166),(163,167),(164,168),(169,173),(170,174),(171,175),(172,176),(177,181),(178,182),(179,183),(180,184),(185,189),(186,190),(187,191),(188,192),(193,197),(194,198),(195,199),(196,200),(201,205),(202,206),(203,207),(204,208),(209,213),(210,214),(211,215),(212,216),(217,221),(218,222),(219,223),(220,224)], [(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)], [(1,148,35,184),(2,151,36,179),(3,146,37,182),(4,149,38,177),(5,152,39,180),(6,147,40,183),(7,150,33,178),(8,145,34,181),(9,63,114,122),(10,58,115,125),(11,61,116,128),(12,64,117,123),(13,59,118,126),(14,62,119,121),(15,57,120,124),(16,60,113,127),(17,206,93,130),(18,201,94,133),(19,204,95,136),(20,207,96,131),(21,202,89,134),(22,205,90,129),(23,208,91,132),(24,203,92,135),(25,214,101,138),(26,209,102,141),(27,212,103,144),(28,215,104,139),(29,210,97,142),(30,213,98,137),(31,216,99,140),(32,211,100,143),(41,186,77,158),(42,189,78,153),(43,192,79,156),(44,187,80,159),(45,190,73,154),(46,185,74,157),(47,188,75,160),(48,191,76,155),(49,194,85,166),(50,197,86,161),(51,200,87,164),(52,195,88,167),(53,198,81,162),(54,193,82,165),(55,196,83,168),(56,199,84,163),(65,222,109,171),(66,217,110,174),(67,220,111,169),(68,223,112,172),(69,218,105,175),(70,221,106,170),(71,224,107,173),(72,219,108,176)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AJ | 28AK | ··· | 28CF | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | D4 | Q8 | D4 | C4○D8 | C7×D4 | C7×Q8 | C7×D4 | C7×C4○D8 |
| kernel | C7×C23.25D4 | C7×C4.Q8 | C7×C2.D8 | C7×C42⋊C2 | C22×C56 | C2×C56 | C23.25D4 | C4.Q8 | C2.D8 | C42⋊C2 | C22×C8 | C2×C8 | C2×C28 | C2×C28 | C22×C14 | C14 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 6 | 12 | 12 | 12 | 6 | 48 | 1 | 2 | 1 | 8 | 6 | 12 | 6 | 48 |
Matrix representation of C7×C23.25D4 ►in GL3(𝔽113) generated by
| 16 | 0 | 0 |
| 0 | 49 | 0 |
| 0 | 0 | 49 |
| 112 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 0 | 112 |
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 1 | 0 | 0 |
| 0 | 18 | 31 |
| 0 | 0 | 69 |
| 98 | 0 | 0 |
| 0 | 18 | 31 |
| 0 | 77 | 95 |
G:=sub<GL(3,GF(113))| [16,0,0,0,49,0,0,0,49],[112,0,0,0,1,0,0,1,112],[112,0,0,0,1,0,0,0,1],[1,0,0,0,112,0,0,0,112],[1,0,0,0,18,0,0,31,69],[98,0,0,0,18,77,0,31,95] >;
C7×C23.25D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._{25}D_4 % in TeX
G:=Group("C7xC2^3.25D4"); // GroupNames label
G:=SmallGroup(448,835);
// by ID
G=gap.SmallGroup(448,835);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,400,1192,9804,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=1,e^4=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations