metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.58(D4×D7), (C22×D7).65D4, C2.C42⋊1D7, C14.C42⋊1C2, D14.9(C22⋊C4), (C22×C4).297D14, C2.8(C42⋊D7), C7⋊1(C23.34D4), C14.4(C42⋊C2), C2.3(D14.D4), C2.2(D14.5D4), C22.33(C4○D28), (C22×C28).12C22, (C23×D7).81C22, C23.254(C22×D7), C22.35(D4⋊2D7), (C22×C14).289C23, C22.16(Q8⋊2D7), C14.8(C22.D4), (C22×Dic7).13C22, (C2×C4×D7)⋊9C4, C2.6(D7×C22⋊C4), C22.88(C2×C4×D7), (C2×D14⋊C4).1C2, (C2×C4).124(C4×D7), C14.3(C2×C22⋊C4), (D7×C22×C4).12C2, (C2×C28).142(C2×C4), C2.7(C4⋊C4⋊7D7), (C2×C14).198(C2×D4), (C2×C14).48(C22×C4), (C2×Dic7).77(C2×C4), (C22×D7).48(C2×C4), (C2×C14).182(C4○D4), (C7×C2.C42)⋊18C2, SmallGroup(448,198)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.58(D4×D7)
G = < a,b,c,d,e,f | a2=b2=c4=e7=f2=1, d2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fcf=bc=cb, bd=db, be=eb, bf=fb, dcd-1=abc-1, ce=ec, de=ed, df=fd, fef=e-1 >
Subgroups: 1116 in 218 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, C14, C14, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2.C42, C2.C42, C2×C22⋊C4, C23×C4, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C23.34D4, D14⋊C4, C2×C4×D7, C2×C4×D7, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C23×D7, C14.C42, C14.C42, C7×C2.C42, C2×D14⋊C4, D7×C22×C4, C22.58(D4×D7)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, C4○D4, D14, C2×C22⋊C4, C42⋊C2, C22.D4, C4×D7, C22×D7, C23.34D4, C2×C4×D7, C4○D28, D4×D7, D4⋊2D7, Q8⋊2D7, C42⋊D7, D7×C22⋊C4, D14.D4, C4⋊C4⋊7D7, D14.5D4, C22.58(D4×D7)
►On 224 points
Generators in S224
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(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)
(1 218 50 169)(2 219 51 170)(3 220 52 171)(4 221 53 172)(5 222 54 173)(6 223 55 174)(7 224 56 175)(8 211 43 176)(9 212 44 177)(10 213 45 178)(11 214 46 179)(12 215 47 180)(13 216 48 181)(14 217 49 182)(15 204 36 183)(16 205 37 184)(17 206 38 185)(18 207 39 186)(19 208 40 187)(20 209 41 188)(21 210 42 189)(22 197 29 190)(23 198 30 191)(24 199 31 192)(25 200 32 193)(26 201 33 194)(27 202 34 195)(28 203 35 196)(57 134 106 141)(58 135 107 142)(59 136 108 143)(60 137 109 144)(61 138 110 145)(62 139 111 146)(63 140 112 147)(64 127 99 148)(65 128 100 149)(66 129 101 150)(67 130 102 151)(68 131 103 152)(69 132 104 153)(70 133 105 154)(71 120 92 155)(72 121 93 156)(73 122 94 157)(74 123 95 158)(75 124 96 159)(76 125 97 160)(77 126 98 161)(78 113 85 162)(79 114 86 163)(80 115 87 164)(81 116 88 165)(82 117 89 166)(83 118 90 167)(84 119 91 168)
(1 127 15 113)(2 128 16 114)(3 129 17 115)(4 130 18 116)(5 131 19 117)(6 132 20 118)(7 133 21 119)(8 134 22 120)(9 135 23 121)(10 136 24 122)(11 137 25 123)(12 138 26 124)(13 139 27 125)(14 140 28 126)(29 155 43 141)(30 156 44 142)(31 157 45 143)(32 158 46 144)(33 159 47 145)(34 160 48 146)(35 161 49 147)(36 162 50 148)(37 163 51 149)(38 164 52 150)(39 165 53 151)(40 166 54 152)(41 167 55 153)(42 168 56 154)(57 183 71 169)(58 184 72 170)(59 185 73 171)(60 186 74 172)(61 187 75 173)(62 188 76 174)(63 189 77 175)(64 190 78 176)(65 191 79 177)(66 192 80 178)(67 193 81 179)(68 194 82 180)(69 195 83 181)(70 196 84 182)(85 211 99 197)(86 212 100 198)(87 213 101 199)(88 214 102 200)(89 215 103 201)(90 216 104 202)(91 217 105 203)(92 218 106 204)(93 219 107 205)(94 220 108 206)(95 221 109 207)(96 222 110 208)(97 223 111 209)(98 224 112 210)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(57 105)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 112)(65 111)(66 110)(67 109)(68 108)(69 107)(70 106)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 98)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)(113 147)(114 146)(115 145)(116 144)(117 143)(118 142)(119 141)(120 154)(121 153)(122 152)(123 151)(124 150)(125 149)(126 148)(127 161)(128 160)(129 159)(130 158)(131 157)(132 156)(133 155)(134 168)(135 167)(136 166)(137 165)(138 164)(139 163)(140 162)(169 217)(170 216)(171 215)(172 214)(173 213)(174 212)(175 211)(176 224)(177 223)(178 222)(179 221)(180 220)(181 219)(182 218)(183 203)(184 202)(185 201)(186 200)(187 199)(188 198)(189 197)(190 210)(191 209)(192 208)(193 207)(194 206)(195 205)(196 204)
G:=sub<Sym(224)| (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(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), (1,218,50,169)(2,219,51,170)(3,220,52,171)(4,221,53,172)(5,222,54,173)(6,223,55,174)(7,224,56,175)(8,211,43,176)(9,212,44,177)(10,213,45,178)(11,214,46,179)(12,215,47,180)(13,216,48,181)(14,217,49,182)(15,204,36,183)(16,205,37,184)(17,206,38,185)(18,207,39,186)(19,208,40,187)(20,209,41,188)(21,210,42,189)(22,197,29,190)(23,198,30,191)(24,199,31,192)(25,200,32,193)(26,201,33,194)(27,202,34,195)(28,203,35,196)(57,134,106,141)(58,135,107,142)(59,136,108,143)(60,137,109,144)(61,138,110,145)(62,139,111,146)(63,140,112,147)(64,127,99,148)(65,128,100,149)(66,129,101,150)(67,130,102,151)(68,131,103,152)(69,132,104,153)(70,133,105,154)(71,120,92,155)(72,121,93,156)(73,122,94,157)(74,123,95,158)(75,124,96,159)(76,125,97,160)(77,126,98,161)(78,113,85,162)(79,114,86,163)(80,115,87,164)(81,116,88,165)(82,117,89,166)(83,118,90,167)(84,119,91,168), (1,127,15,113)(2,128,16,114)(3,129,17,115)(4,130,18,116)(5,131,19,117)(6,132,20,118)(7,133,21,119)(8,134,22,120)(9,135,23,121)(10,136,24,122)(11,137,25,123)(12,138,26,124)(13,139,27,125)(14,140,28,126)(29,155,43,141)(30,156,44,142)(31,157,45,143)(32,158,46,144)(33,159,47,145)(34,160,48,146)(35,161,49,147)(36,162,50,148)(37,163,51,149)(38,164,52,150)(39,165,53,151)(40,166,54,152)(41,167,55,153)(42,168,56,154)(57,183,71,169)(58,184,72,170)(59,185,73,171)(60,186,74,172)(61,187,75,173)(62,188,76,174)(63,189,77,175)(64,190,78,176)(65,191,79,177)(66,192,80,178)(67,193,81,179)(68,194,82,180)(69,195,83,181)(70,196,84,182)(85,211,99,197)(86,212,100,198)(87,213,101,199)(88,214,102,200)(89,215,103,201)(90,216,104,202)(91,217,105,203)(92,218,106,204)(93,219,107,205)(94,220,108,206)(95,221,109,207)(96,222,110,208)(97,223,111,209)(98,224,112,210), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,112)(65,111)(66,110)(67,109)(68,108)(69,107)(70,106)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,154)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,161)(128,160)(129,159)(130,158)(131,157)(132,156)(133,155)(134,168)(135,167)(136,166)(137,165)(138,164)(139,163)(140,162)(169,217)(170,216)(171,215)(172,214)(173,213)(174,212)(175,211)(176,224)(177,223)(178,222)(179,221)(180,220)(181,219)(182,218)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,210)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(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), (1,218,50,169)(2,219,51,170)(3,220,52,171)(4,221,53,172)(5,222,54,173)(6,223,55,174)(7,224,56,175)(8,211,43,176)(9,212,44,177)(10,213,45,178)(11,214,46,179)(12,215,47,180)(13,216,48,181)(14,217,49,182)(15,204,36,183)(16,205,37,184)(17,206,38,185)(18,207,39,186)(19,208,40,187)(20,209,41,188)(21,210,42,189)(22,197,29,190)(23,198,30,191)(24,199,31,192)(25,200,32,193)(26,201,33,194)(27,202,34,195)(28,203,35,196)(57,134,106,141)(58,135,107,142)(59,136,108,143)(60,137,109,144)(61,138,110,145)(62,139,111,146)(63,140,112,147)(64,127,99,148)(65,128,100,149)(66,129,101,150)(67,130,102,151)(68,131,103,152)(69,132,104,153)(70,133,105,154)(71,120,92,155)(72,121,93,156)(73,122,94,157)(74,123,95,158)(75,124,96,159)(76,125,97,160)(77,126,98,161)(78,113,85,162)(79,114,86,163)(80,115,87,164)(81,116,88,165)(82,117,89,166)(83,118,90,167)(84,119,91,168), (1,127,15,113)(2,128,16,114)(3,129,17,115)(4,130,18,116)(5,131,19,117)(6,132,20,118)(7,133,21,119)(8,134,22,120)(9,135,23,121)(10,136,24,122)(11,137,25,123)(12,138,26,124)(13,139,27,125)(14,140,28,126)(29,155,43,141)(30,156,44,142)(31,157,45,143)(32,158,46,144)(33,159,47,145)(34,160,48,146)(35,161,49,147)(36,162,50,148)(37,163,51,149)(38,164,52,150)(39,165,53,151)(40,166,54,152)(41,167,55,153)(42,168,56,154)(57,183,71,169)(58,184,72,170)(59,185,73,171)(60,186,74,172)(61,187,75,173)(62,188,76,174)(63,189,77,175)(64,190,78,176)(65,191,79,177)(66,192,80,178)(67,193,81,179)(68,194,82,180)(69,195,83,181)(70,196,84,182)(85,211,99,197)(86,212,100,198)(87,213,101,199)(88,214,102,200)(89,215,103,201)(90,216,104,202)(91,217,105,203)(92,218,106,204)(93,219,107,205)(94,220,108,206)(95,221,109,207)(96,222,110,208)(97,223,111,209)(98,224,112,210), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,112)(65,111)(66,110)(67,109)(68,108)(69,107)(70,106)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,154)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,161)(128,160)(129,159)(130,158)(131,157)(132,156)(133,155)(134,168)(135,167)(136,166)(137,165)(138,164)(139,163)(140,162)(169,217)(170,216)(171,215)(172,214)(173,213)(174,212)(175,211)(176,224)(177,223)(178,222)(179,221)(180,220)(181,219)(182,218)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,210)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204) );
G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208),(181,209),(182,210),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(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)], [(1,218,50,169),(2,219,51,170),(3,220,52,171),(4,221,53,172),(5,222,54,173),(6,223,55,174),(7,224,56,175),(8,211,43,176),(9,212,44,177),(10,213,45,178),(11,214,46,179),(12,215,47,180),(13,216,48,181),(14,217,49,182),(15,204,36,183),(16,205,37,184),(17,206,38,185),(18,207,39,186),(19,208,40,187),(20,209,41,188),(21,210,42,189),(22,197,29,190),(23,198,30,191),(24,199,31,192),(25,200,32,193),(26,201,33,194),(27,202,34,195),(28,203,35,196),(57,134,106,141),(58,135,107,142),(59,136,108,143),(60,137,109,144),(61,138,110,145),(62,139,111,146),(63,140,112,147),(64,127,99,148),(65,128,100,149),(66,129,101,150),(67,130,102,151),(68,131,103,152),(69,132,104,153),(70,133,105,154),(71,120,92,155),(72,121,93,156),(73,122,94,157),(74,123,95,158),(75,124,96,159),(76,125,97,160),(77,126,98,161),(78,113,85,162),(79,114,86,163),(80,115,87,164),(81,116,88,165),(82,117,89,166),(83,118,90,167),(84,119,91,168)], [(1,127,15,113),(2,128,16,114),(3,129,17,115),(4,130,18,116),(5,131,19,117),(6,132,20,118),(7,133,21,119),(8,134,22,120),(9,135,23,121),(10,136,24,122),(11,137,25,123),(12,138,26,124),(13,139,27,125),(14,140,28,126),(29,155,43,141),(30,156,44,142),(31,157,45,143),(32,158,46,144),(33,159,47,145),(34,160,48,146),(35,161,49,147),(36,162,50,148),(37,163,51,149),(38,164,52,150),(39,165,53,151),(40,166,54,152),(41,167,55,153),(42,168,56,154),(57,183,71,169),(58,184,72,170),(59,185,73,171),(60,186,74,172),(61,187,75,173),(62,188,76,174),(63,189,77,175),(64,190,78,176),(65,191,79,177),(66,192,80,178),(67,193,81,179),(68,194,82,180),(69,195,83,181),(70,196,84,182),(85,211,99,197),(86,212,100,198),(87,213,101,199),(88,214,102,200),(89,215,103,201),(90,216,104,202),(91,217,105,203),(92,218,106,204),(93,219,107,205),(94,220,108,206),(95,221,109,207),(96,222,110,208),(97,223,111,209),(98,224,112,210)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(57,105),(58,104),(59,103),(60,102),(61,101),(62,100),(63,99),(64,112),(65,111),(66,110),(67,109),(68,108),(69,107),(70,106),(71,91),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,98),(79,97),(80,96),(81,95),(82,94),(83,93),(84,92),(113,147),(114,146),(115,145),(116,144),(117,143),(118,142),(119,141),(120,154),(121,153),(122,152),(123,151),(124,150),(125,149),(126,148),(127,161),(128,160),(129,159),(130,158),(131,157),(132,156),(133,155),(134,168),(135,167),(136,166),(137,165),(138,164),(139,163),(140,162),(169,217),(170,216),(171,215),(172,214),(173,213),(174,212),(175,211),(176,224),(177,223),(178,222),(179,221),(180,220),(181,219),(182,218),(183,203),(184,202),(185,201),(186,200),(187,199),(188,198),(189,197),(190,210),(191,209),(192,208),(193,207),(194,206),(195,205),(196,204)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | C4×D7 | C4○D28 | D4×D7 | D4⋊2D7 | Q8⋊2D7 |
| kernel | C22.58(D4×D7) | C14.C42 | C7×C2.C42 | C2×D14⋊C4 | D7×C22×C4 | C2×C4×D7 | C22×D7 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 1 | 2 | 1 | 8 | 4 | 3 | 8 | 9 | 12 | 24 | 6 | 3 | 3 |
Matrix representation of C22.58(D4×D7) ►in GL6(𝔽29)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 14 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 24 |
| 0 | 0 | 0 | 0 | 28 | 5 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 5 | 17 |
| 3 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 28 | 0 | 0 |
| 0 | 0 | 2 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 26 | 1 | 0 | 0 | 0 | 0 |
| 21 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 19 | 0 | 0 |
| 0 | 0 | 8 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 27 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,25,1,0,0,0,0,14,4,0,0,0,0,0,0,24,28,0,0,0,0,24,5],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,5,0,0,0,0,0,17],[3,1,0,0,0,0,28,0,0,0,0,0,0,0,10,2,0,0,0,0,28,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,21,0,0,0,0,1,3,0,0,0,0,0,0,20,8,0,0,0,0,19,9,0,0,0,0,0,0,1,27,0,0,0,0,0,28] >;
C22.58(D4×D7) in GAP, Magma, Sage, TeX
C_2^2._{58}(D_4\times D_7) % in TeX
G:=Group("C2^2.58(D4xD7)"); // GroupNames label
G:=SmallGroup(448,198);
// by ID
G=gap.SmallGroup(448,198);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,422,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=e^7=f^2=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*c*f=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=a*b*c^-1,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations