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G = C23.8D30order 480 = 25·3·5

3rd non-split extension by C23 of D30 acting via D30/C15=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.8D30, C605C49C2, (C2×C4).26D30, (C2×C12).32D10, (C2×C20).206D6, C22⋊C4.2D15, C30.4Q85C2, (C4×Dic15)⋊16C2, C6.95(C4○D20), (C2×C60).17C22, (C22×C6).54D10, (C22×C10).69D6, C57(C23.8D6), C1518(C422C2), C30.215(C4○D4), C10.95(C4○D12), C6.90(D42D5), C2.7(D42D15), (C2×C30).276C23, C30.38D4.3C2, C37(C23.D10), C10.90(D42S3), C2.9(D6011C2), (C22×C30).10C22, (C2×Dic15).6C22, C22.40(C22×D15), (C5×C22⋊C4).2S3, (C3×C22⋊C4).2D5, (C15×C22⋊C4).2C2, (C2×C6).272(C22×D5), (C2×C10).271(C22×S3), SmallGroup(480,844)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C23.8D30
C1C5C15C30C2×C30C2×Dic15C4×Dic15 — C23.8D30
C15C2×C30 — C23.8D30
C1C22C22⋊C4

Generators and relations for C23.8D30
 G = < a,b,c,d,e | a2=b2=c2=1, d30=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d29 >

Subgroups: 548 in 120 conjugacy classes, 47 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C23, C10 [×3], C10, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×3], C15, C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×3], C2×Dic3 [×4], C2×C12 [×2], C22×C6, C30 [×3], C30, C422C2, C2×Dic5 [×4], C2×C20 [×2], C22×C10, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C3×C22⋊C4, Dic15 [×4], C60 [×2], C2×C30, C2×C30 [×3], C4×Dic5, C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C5×C22⋊C4, C23.8D6, C2×Dic15 [×4], C2×C60 [×2], C22×C30, C23.D10, C4×Dic15, C30.4Q8 [×2], C605C4, C30.38D4 [×2], C15×C22⋊C4, C23.8D30
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4 [×3], D10 [×3], C22×S3, D15, C422C2, C22×D5, C4○D12, D42S3 [×2], D30 [×3], C4○D20, D42D5 [×2], C23.8D6, C22×D15, C23.D10, D6011C2, D42D15 [×2], C23.8D30

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

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

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

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

84 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B15C15D20A···20H30A···30L30M···30T60A···60P
order122223444444444556666610···1010101010121212121515151520···2030···3030···3060···60
size11114222430303030606022222442···24444444422224···42···24···44···4

84 irreducible representations

dim1111112222222222222444
type+++++++++++++++---
imageC1C2C2C2C2C2S3D5D6D6C4○D4D10D10D15C4○D12D30D30C4○D20D6011C2D42S3D42D5D42D15
kernelC23.8D30C4×Dic15C30.4Q8C605C4C30.38D4C15×C22⋊C4C5×C22⋊C4C3×C22⋊C4C2×C20C22×C10C30C2×C12C22×C6C22⋊C4C10C2×C4C23C6C2C10C6C2
# reps11212112216424484816248

Matrix representation of C23.8D30 in GL6(𝔽61)

100000
010000
001000
0016000
000010
00004160
,
100000
010000
0060000
0006000
000010
000001
,
100000
010000
001000
000100
0000600
0000060
,
4600000
2140000
0050000
0005000
00004253
00004519
,
4530000
40570000
00502200
0001100
0000500
0000050

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,60,0,0,0,0,0,0,1,41,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[46,21,0,0,0,0,0,4,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,42,45,0,0,0,0,53,19],[4,40,0,0,0,0,53,57,0,0,0,0,0,0,50,0,0,0,0,0,22,11,0,0,0,0,0,0,50,0,0,0,0,0,0,50] >;

C23.8D30 in GAP, Magma, Sage, TeX

C_2^3._8D_{30}
% in TeX

G:=Group("C2^3.8D30");
// GroupNames label

G:=SmallGroup(480,844);
// by ID

G=gap.SmallGroup(480,844);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,590,219,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^30=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-1=a*c=c*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