Copied to
clipboard

G = C23.26D30order 480 = 25·3·5

2nd non-split extension by C23 of D30 acting via D30/C30=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.26D30, (C2×C60)⋊20C4, C605C435C2, (C2×C4)⋊4Dic15, (C2×C12)⋊6Dic5, C60.239(C2×C4), (C2×C20)⋊11Dic3, (C2×C20).417D6, (C2×C4).103D30, (C22×C4).9D15, (C4×Dic15)⋊21C2, (C2×C12).400D10, (C22×C20).14S3, (C22×C60).13C2, C20.65(C2×Dic3), C12.44(C2×Dic5), C4.15(C2×Dic15), (C22×C12).10D5, C1530(C42⋊C2), C6.104(C4○D20), C30.176(C4○D4), C30.214(C22×C4), (C2×C60).481C22, (C2×C30).300C23, (C22×C10).135D6, (C22×C6).117D10, C55(C23.26D6), C10.104(C4○D12), C2.4(D6011C2), C22.5(C2×Dic15), C6.25(C22×Dic5), C2.5(C22×Dic15), C30.38D4.11C2, C34(C23.21D10), C10.38(C22×Dic3), C22.22(C22×D15), (C22×C30).140C22, (C2×Dic15).169C22, (C2×C30).180(C2×C4), (C2×C6).37(C2×Dic5), (C2×C10).57(C2×Dic3), (C2×C6).296(C22×D5), (C2×C10).295(C22×S3), SmallGroup(480,891)

Series: Derived Chief Lower central Upper central

C1C30 — C23.26D30
C1C5C15C30C2×C30C2×Dic15C4×Dic15 — C23.26D30
C15C30 — C23.26D30
C1C2×C4C22×C4

Generators and relations for C23.26D30
 G = < a,b,c,d,e | a2=b2=c2=1, d30=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d29 >

Subgroups: 564 in 152 conjugacy classes, 87 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C5, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C22×C6, C30, C30 [×2], C30 [×2], C42⋊C2, C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C4×Dic3 [×2], C4⋊Dic3 [×2], C6.D4 [×2], C22×C12, Dic15 [×4], C60 [×4], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×2], C22×C20, C23.26D6, C2×Dic15 [×4], C2×C60 [×2], C2×C60 [×4], C22×C30, C23.21D10, C4×Dic15 [×2], C605C4 [×2], C30.38D4 [×2], C22×C60, C23.26D30
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, Dic3 [×4], D6 [×3], C22×C4, C4○D4 [×2], Dic5 [×4], D10 [×3], C2×Dic3 [×6], C22×S3, D15, C42⋊C2, C2×Dic5 [×6], C22×D5, C4○D12 [×2], C22×Dic3, Dic15 [×4], D30 [×3], C4○D20 [×2], C22×Dic5, C23.26D6, C2×Dic15 [×6], C22×D15, C23.21D10, D6011C2 [×2], C22×Dic15, C23.26D30

Smallest permutation representation of C23.26D30
On 240 points
Generators in S240
(1 157)(2 158)(3 159)(4 160)(5 161)(6 162)(7 163)(8 164)(9 165)(10 166)(11 167)(12 168)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 153)(58 154)(59 155)(60 156)(61 229)(62 230)(63 231)(64 232)(65 233)(66 234)(67 235)(68 236)(69 237)(70 238)(71 239)(72 240)(73 181)(74 182)(75 183)(76 184)(77 185)(78 186)(79 187)(80 188)(81 189)(82 190)(83 191)(84 192)(85 193)(86 194)(87 195)(88 196)(89 197)(90 198)(91 199)(92 200)(93 201)(94 202)(95 203)(96 204)(97 205)(98 206)(99 207)(100 208)(101 209)(102 210)(103 211)(104 212)(105 213)(106 214)(107 215)(108 216)(109 217)(110 218)(111 219)(112 220)(113 221)(114 222)(115 223)(116 224)(117 225)(118 226)(119 227)(120 228)
(1 157)(2 158)(3 159)(4 160)(5 161)(6 162)(7 163)(8 164)(9 165)(10 166)(11 167)(12 168)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 153)(58 154)(59 155)(60 156)(61 199)(62 200)(63 201)(64 202)(65 203)(66 204)(67 205)(68 206)(69 207)(70 208)(71 209)(72 210)(73 211)(74 212)(75 213)(76 214)(77 215)(78 216)(79 217)(80 218)(81 219)(82 220)(83 221)(84 222)(85 223)(86 224)(87 225)(88 226)(89 227)(90 228)(91 229)(92 230)(93 231)(94 232)(95 233)(96 234)(97 235)(98 236)(99 237)(100 238)(101 239)(102 240)(103 181)(104 182)(105 183)(106 184)(107 185)(108 186)(109 187)(110 188)(111 189)(112 190)(113 191)(114 192)(115 193)(116 194)(117 195)(118 196)(119 197)(120 198)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 169)(140 170)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)(181 211)(182 212)(183 213)(184 214)(185 215)(186 216)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(199 229)(200 230)(201 231)(202 232)(203 233)(204 234)(205 235)(206 236)(207 237)(208 238)(209 239)(210 240)
(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)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 201 127 93)(2 230 128 62)(3 199 129 91)(4 228 130 120)(5 197 131 89)(6 226 132 118)(7 195 133 87)(8 224 134 116)(9 193 135 85)(10 222 136 114)(11 191 137 83)(12 220 138 112)(13 189 139 81)(14 218 140 110)(15 187 141 79)(16 216 142 108)(17 185 143 77)(18 214 144 106)(19 183 145 75)(20 212 146 104)(21 181 147 73)(22 210 148 102)(23 239 149 71)(24 208 150 100)(25 237 151 69)(26 206 152 98)(27 235 153 67)(28 204 154 96)(29 233 155 65)(30 202 156 94)(31 231 157 63)(32 200 158 92)(33 229 159 61)(34 198 160 90)(35 227 161 119)(36 196 162 88)(37 225 163 117)(38 194 164 86)(39 223 165 115)(40 192 166 84)(41 221 167 113)(42 190 168 82)(43 219 169 111)(44 188 170 80)(45 217 171 109)(46 186 172 78)(47 215 173 107)(48 184 174 76)(49 213 175 105)(50 182 176 74)(51 211 177 103)(52 240 178 72)(53 209 179 101)(54 238 180 70)(55 207 121 99)(56 236 122 68)(57 205 123 97)(58 234 124 66)(59 203 125 95)(60 232 126 64)

G:=sub<Sym(240)| (1,157)(2,158)(3,159)(4,160)(5,161)(6,162)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,229)(62,230)(63,231)(64,232)(65,233)(66,234)(67,235)(68,236)(69,237)(70,238)(71,239)(72,240)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,189)(82,190)(83,191)(84,192)(85,193)(86,194)(87,195)(88,196)(89,197)(90,198)(91,199)(92,200)(93,201)(94,202)(95,203)(96,204)(97,205)(98,206)(99,207)(100,208)(101,209)(102,210)(103,211)(104,212)(105,213)(106,214)(107,215)(108,216)(109,217)(110,218)(111,219)(112,220)(113,221)(114,222)(115,223)(116,224)(117,225)(118,226)(119,227)(120,228), (1,157)(2,158)(3,159)(4,160)(5,161)(6,162)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,199)(62,200)(63,201)(64,202)(65,203)(66,204)(67,205)(68,206)(69,207)(70,208)(71,209)(72,210)(73,211)(74,212)(75,213)(76,214)(77,215)(78,216)(79,217)(80,218)(81,219)(82,220)(83,221)(84,222)(85,223)(86,224)(87,225)(88,226)(89,227)(90,228)(91,229)(92,230)(93,231)(94,232)(95,233)(96,234)(97,235)(98,236)(99,237)(100,238)(101,239)(102,240)(103,181)(104,182)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)(116,194)(117,195)(118,196)(119,197)(120,198), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,201,127,93)(2,230,128,62)(3,199,129,91)(4,228,130,120)(5,197,131,89)(6,226,132,118)(7,195,133,87)(8,224,134,116)(9,193,135,85)(10,222,136,114)(11,191,137,83)(12,220,138,112)(13,189,139,81)(14,218,140,110)(15,187,141,79)(16,216,142,108)(17,185,143,77)(18,214,144,106)(19,183,145,75)(20,212,146,104)(21,181,147,73)(22,210,148,102)(23,239,149,71)(24,208,150,100)(25,237,151,69)(26,206,152,98)(27,235,153,67)(28,204,154,96)(29,233,155,65)(30,202,156,94)(31,231,157,63)(32,200,158,92)(33,229,159,61)(34,198,160,90)(35,227,161,119)(36,196,162,88)(37,225,163,117)(38,194,164,86)(39,223,165,115)(40,192,166,84)(41,221,167,113)(42,190,168,82)(43,219,169,111)(44,188,170,80)(45,217,171,109)(46,186,172,78)(47,215,173,107)(48,184,174,76)(49,213,175,105)(50,182,176,74)(51,211,177,103)(52,240,178,72)(53,209,179,101)(54,238,180,70)(55,207,121,99)(56,236,122,68)(57,205,123,97)(58,234,124,66)(59,203,125,95)(60,232,126,64)>;

G:=Group( (1,157)(2,158)(3,159)(4,160)(5,161)(6,162)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,229)(62,230)(63,231)(64,232)(65,233)(66,234)(67,235)(68,236)(69,237)(70,238)(71,239)(72,240)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,189)(82,190)(83,191)(84,192)(85,193)(86,194)(87,195)(88,196)(89,197)(90,198)(91,199)(92,200)(93,201)(94,202)(95,203)(96,204)(97,205)(98,206)(99,207)(100,208)(101,209)(102,210)(103,211)(104,212)(105,213)(106,214)(107,215)(108,216)(109,217)(110,218)(111,219)(112,220)(113,221)(114,222)(115,223)(116,224)(117,225)(118,226)(119,227)(120,228), (1,157)(2,158)(3,159)(4,160)(5,161)(6,162)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,153)(58,154)(59,155)(60,156)(61,199)(62,200)(63,201)(64,202)(65,203)(66,204)(67,205)(68,206)(69,207)(70,208)(71,209)(72,210)(73,211)(74,212)(75,213)(76,214)(77,215)(78,216)(79,217)(80,218)(81,219)(82,220)(83,221)(84,222)(85,223)(86,224)(87,225)(88,226)(89,227)(90,228)(91,229)(92,230)(93,231)(94,232)(95,233)(96,234)(97,235)(98,236)(99,237)(100,238)(101,239)(102,240)(103,181)(104,182)(105,183)(106,184)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(113,191)(114,192)(115,193)(116,194)(117,195)(118,196)(119,197)(120,198), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240), (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)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,201,127,93)(2,230,128,62)(3,199,129,91)(4,228,130,120)(5,197,131,89)(6,226,132,118)(7,195,133,87)(8,224,134,116)(9,193,135,85)(10,222,136,114)(11,191,137,83)(12,220,138,112)(13,189,139,81)(14,218,140,110)(15,187,141,79)(16,216,142,108)(17,185,143,77)(18,214,144,106)(19,183,145,75)(20,212,146,104)(21,181,147,73)(22,210,148,102)(23,239,149,71)(24,208,150,100)(25,237,151,69)(26,206,152,98)(27,235,153,67)(28,204,154,96)(29,233,155,65)(30,202,156,94)(31,231,157,63)(32,200,158,92)(33,229,159,61)(34,198,160,90)(35,227,161,119)(36,196,162,88)(37,225,163,117)(38,194,164,86)(39,223,165,115)(40,192,166,84)(41,221,167,113)(42,190,168,82)(43,219,169,111)(44,188,170,80)(45,217,171,109)(46,186,172,78)(47,215,173,107)(48,184,174,76)(49,213,175,105)(50,182,176,74)(51,211,177,103)(52,240,178,72)(53,209,179,101)(54,238,180,70)(55,207,121,99)(56,236,122,68)(57,205,123,97)(58,234,124,66)(59,203,125,95)(60,232,126,64) );

G=PermutationGroup([(1,157),(2,158),(3,159),(4,160),(5,161),(6,162),(7,163),(8,164),(9,165),(10,166),(11,167),(12,168),(13,169),(14,170),(15,171),(16,172),(17,173),(18,174),(19,175),(20,176),(21,177),(22,178),(23,179),(24,180),(25,121),(26,122),(27,123),(28,124),(29,125),(30,126),(31,127),(32,128),(33,129),(34,130),(35,131),(36,132),(37,133),(38,134),(39,135),(40,136),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(57,153),(58,154),(59,155),(60,156),(61,229),(62,230),(63,231),(64,232),(65,233),(66,234),(67,235),(68,236),(69,237),(70,238),(71,239),(72,240),(73,181),(74,182),(75,183),(76,184),(77,185),(78,186),(79,187),(80,188),(81,189),(82,190),(83,191),(84,192),(85,193),(86,194),(87,195),(88,196),(89,197),(90,198),(91,199),(92,200),(93,201),(94,202),(95,203),(96,204),(97,205),(98,206),(99,207),(100,208),(101,209),(102,210),(103,211),(104,212),(105,213),(106,214),(107,215),(108,216),(109,217),(110,218),(111,219),(112,220),(113,221),(114,222),(115,223),(116,224),(117,225),(118,226),(119,227),(120,228)], [(1,157),(2,158),(3,159),(4,160),(5,161),(6,162),(7,163),(8,164),(9,165),(10,166),(11,167),(12,168),(13,169),(14,170),(15,171),(16,172),(17,173),(18,174),(19,175),(20,176),(21,177),(22,178),(23,179),(24,180),(25,121),(26,122),(27,123),(28,124),(29,125),(30,126),(31,127),(32,128),(33,129),(34,130),(35,131),(36,132),(37,133),(38,134),(39,135),(40,136),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(57,153),(58,154),(59,155),(60,156),(61,199),(62,200),(63,201),(64,202),(65,203),(66,204),(67,205),(68,206),(69,207),(70,208),(71,209),(72,210),(73,211),(74,212),(75,213),(76,214),(77,215),(78,216),(79,217),(80,218),(81,219),(82,220),(83,221),(84,222),(85,223),(86,224),(87,225),(88,226),(89,227),(90,228),(91,229),(92,230),(93,231),(94,232),(95,233),(96,234),(97,235),(98,236),(99,237),(100,238),(101,239),(102,240),(103,181),(104,182),(105,183),(106,184),(107,185),(108,186),(109,187),(110,188),(111,189),(112,190),(113,191),(114,192),(115,193),(116,194),(117,195),(118,196),(119,197),(120,198)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120),(121,151),(122,152),(123,153),(124,154),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160),(131,161),(132,162),(133,163),(134,164),(135,165),(136,166),(137,167),(138,168),(139,169),(140,170),(141,171),(142,172),(143,173),(144,174),(145,175),(146,176),(147,177),(148,178),(149,179),(150,180),(181,211),(182,212),(183,213),(184,214),(185,215),(186,216),(187,217),(188,218),(189,219),(190,220),(191,221),(192,222),(193,223),(194,224),(195,225),(196,226),(197,227),(198,228),(199,229),(200,230),(201,231),(202,232),(203,233),(204,234),(205,235),(206,236),(207,237),(208,238),(209,239),(210,240)], [(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),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,201,127,93),(2,230,128,62),(3,199,129,91),(4,228,130,120),(5,197,131,89),(6,226,132,118),(7,195,133,87),(8,224,134,116),(9,193,135,85),(10,222,136,114),(11,191,137,83),(12,220,138,112),(13,189,139,81),(14,218,140,110),(15,187,141,79),(16,216,142,108),(17,185,143,77),(18,214,144,106),(19,183,145,75),(20,212,146,104),(21,181,147,73),(22,210,148,102),(23,239,149,71),(24,208,150,100),(25,237,151,69),(26,206,152,98),(27,235,153,67),(28,204,154,96),(29,233,155,65),(30,202,156,94),(31,231,157,63),(32,200,158,92),(33,229,159,61),(34,198,160,90),(35,227,161,119),(36,196,162,88),(37,225,163,117),(38,194,164,86),(39,223,165,115),(40,192,166,84),(41,221,167,113),(42,190,168,82),(43,219,169,111),(44,188,170,80),(45,217,171,109),(46,186,172,78),(47,215,173,107),(48,184,174,76),(49,213,175,105),(50,182,176,74),(51,211,177,103),(52,240,178,72),(53,209,179,101),(54,238,180,70),(55,207,121,99),(56,236,122,68),(57,205,123,97),(58,234,124,66),(59,203,125,95),(60,232,126,64)])

132 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G···4N5A5B6A···6G10A···10N12A···12H15A15B15C15D20A···20P30A···30AB60A···60AF
order12222234444444···4556···610···1012···121515151520···2030···3060···60
size111122211112230···30222···22···22···222222···22···22···2

132 irreducible representations

dim1111112222222222222222
type+++++++-++-+++-++
imageC1C2C2C2C2C4S3D5Dic3D6D6C4○D4Dic5D10D10D15C4○D12Dic15D30D30C4○D20D6011C2
kernelC23.26D30C4×Dic15C605C4C30.38D4C22×C60C2×C60C22×C20C22×C12C2×C20C2×C20C22×C10C30C2×C12C2×C12C22×C6C22×C4C10C2×C4C2×C4C23C6C2
# reps1222181242148424816841632

Matrix representation of C23.26D30 in GL3(𝔽61) generated by

100
0600
001
,
6000
0600
0060
,
100
0600
0060
,
6000
0310
0059
,
1100
0059
0300
G:=sub<GL(3,GF(61))| [1,0,0,0,60,0,0,0,1],[60,0,0,0,60,0,0,0,60],[1,0,0,0,60,0,0,0,60],[60,0,0,0,31,0,0,0,59],[11,0,0,0,0,30,0,59,0] >;

C23.26D30 in GAP, Magma, Sage, TeX

C_2^3._{26}D_{30}
% in TeX

G:=Group("C2^3.26D30");
// GroupNames label

G:=SmallGroup(480,891);
// by ID

G=gap.SmallGroup(480,891);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^30=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^29>;
// generators/relations

׿
×
𝔽