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G = C23.15D30order 480 = 25·3·5

1st non-split extension by C23 of D30 acting via D30/D15=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.15D30, (C2×C4).25D30, (C2×Dic15)⋊9C4, (C2×C20).205D6, C22⋊C4.3D15, C30.4Q88C2, C22.6(C4×D15), (C4×Dic15)⋊15C2, (C2×C12).203D10, (C22×C10).67D6, (C22×C6).52D10, C1524(C42⋊C2), C30.213(C4○D4), C2.1(D42D15), C6.88(D42D5), (C2×C30).274C23, C30.156(C22×C4), (C2×C60).171C22, Dic15.42(C2×C4), C30.38D4.1C2, C55(C23.16D6), (C22×C30).8C22, C10.88(D42S3), C34(C23.11D10), (C22×Dic15).2C2, C22.12(C22×D15), (C2×Dic15).237C22, C6.61(C2×C4×D5), C2.7(C2×C4×D15), C10.93(S3×C2×C4), (C2×C6).12(C4×D5), (C2×C30).67(C2×C4), (C2×C10).35(C4×S3), (C5×C22⋊C4).3S3, (C3×C22⋊C4).3D5, (C15×C22⋊C4).3C2, (C2×C6).270(C22×D5), (C2×C10).269(C22×S3), SmallGroup(480,842)

Series: Derived Chief Lower central Upper central

C1C30 — C23.15D30
C1C5C15C30C2×C30C2×Dic15C22×Dic15 — C23.15D30
C15C30 — C23.15D30
C1C22C22⋊C4

Generators and relations for C23.15D30
 G = < a,b,c,d,e | a2=b2=c2=1, d30=b, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d29 >

Subgroups: 644 in 152 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×2], C5, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], Dic3 [×6], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×2], C22×C4, Dic5 [×6], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×8], C2×C12 [×2], C22×C6, C30, C30 [×2], C30 [×2], C42⋊C2, C2×Dic5 [×8], C2×C20 [×2], C22×C10, C4×Dic3 [×2], Dic3⋊C4 [×2], C6.D4, C3×C22⋊C4, C22×Dic3, Dic15 [×4], Dic15 [×2], C60 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5 [×2], C10.D4 [×2], C23.D5, C5×C22⋊C4, C22×Dic5, C23.16D6, C2×Dic15 [×2], C2×Dic15 [×6], C2×C60 [×2], C22×C30, C23.11D10, C4×Dic15 [×2], C30.4Q8 [×2], C30.38D4, C15×C22⋊C4, C22×Dic15, C23.15D30
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, C4○D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, D15, C42⋊C2, C4×D5 [×2], C22×D5, S3×C2×C4, D42S3 [×2], D30 [×3], C2×C4×D5, D42D5 [×2], C23.16D6, C4×D15 [×2], C22×D15, C23.11D10, C2×C4×D15, D42D15 [×2], C23.15D30

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

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I···4N5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B15C15D20A···20H30A···30L30M···30T60A···60P
order1222223444444444···4556666610···1010101010121212121515151520···2030···3030···3060···60
size111122222221515151530···3022222442···24444444422224···42···24···44···4

90 irreducible representations

dim11111112222222222222444
type+++++++++++++++---
imageC1C2C2C2C2C2C4S3D5D6D6C4○D4D10D10C4×S3D15C4×D5D30D30C4×D15D42S3D42D5D42D15
kernelC23.15D30C4×Dic15C30.4Q8C30.38D4C15×C22⋊C4C22×Dic15C2×Dic15C5×C22⋊C4C3×C22⋊C4C2×C20C22×C10C30C2×C12C22×C6C2×C10C22⋊C4C2×C6C2×C4C23C22C10C6C2
# reps122111812214424488416248

Matrix representation of C23.15D30 in GL4(𝔽61) generated by

60000
06000
0010
00060
,
60000
06000
0010
0001
,
1000
0100
00600
00060
,
553100
213400
0001
0010
,
72500
595400
00050
00500
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[55,21,0,0,31,34,0,0,0,0,0,1,0,0,1,0],[7,59,0,0,25,54,0,0,0,0,0,50,0,0,50,0] >;

C23.15D30 in GAP, Magma, Sage, TeX

C_2^3._{15}D_{30}
% in TeX

G:=Group("C2^3.15D30");
// GroupNames label

G:=SmallGroup(480,842);
// by ID

G=gap.SmallGroup(480,842);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,219,58,2693,18822]);
// Polycyclic

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

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