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G = C5×C123D4order 480 = 25·3·5

Direct product of C5 and C123D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C123D4, C6021D4, C123(C5×D4), (C6×D4)⋊4C10, (C2×D12)⋊9C10, (D4×C10)⋊13S3, (D4×C30)⋊18C2, C209(C3⋊D4), Dic31(C5×D4), C6.52(D4×C10), (C10×D12)⋊25C2, C1510(C41D4), (C5×Dic3)⋊10D4, (C4×Dic3)⋊6C10, C10.195(S3×D4), (C2×C20).363D6, C30.435(C2×D4), (Dic3×C20)⋊18C2, C23.15(S3×C10), (C22×C10).26D6, (C2×C60).363C22, (C2×C30).434C23, (C22×C30).127C22, (C10×Dic3).231C22, C41(C5×C3⋊D4), C32(C5×C41D4), C2.28(C5×S3×D4), (C2×D4)⋊6(C5×S3), (C2×C3⋊D4)⋊7C10, (C10×C3⋊D4)⋊22C2, (C2×C4).52(S3×C10), C2.16(C10×C3⋊D4), C22.62(S3×C2×C10), (C2×C12).36(C2×C10), C10.137(C2×C3⋊D4), (S3×C2×C10).73C22, (C22×C6).22(C2×C10), (C2×C6).55(C22×C10), (C22×S3).12(C2×C10), (C2×C10).368(C22×S3), (C2×Dic3).39(C2×C10), SmallGroup(480,819)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×C123D4
C1C3C6C2×C6C2×C30S3×C2×C10C10×D12 — C5×C123D4
C3C2×C6 — C5×C123D4
C1C2×C10D4×C10

Generators and relations for C5×C123D4
 G = < a,b,c,d | a5=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 580 in 216 conjugacy classes, 74 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×12], C5, S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×2], D4 [×12], C23 [×2], C23 [×2], C10, C10 [×2], C10 [×4], Dic3 [×4], C12 [×2], D6 [×6], C2×C6, C2×C6 [×6], C15, C42, C2×D4, C2×D4 [×5], C20 [×2], C20 [×4], C2×C10, C2×C10 [×12], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C5×S3 [×2], C30, C30 [×2], C30 [×2], C41D4, C2×C20, C2×C20 [×2], C5×D4 [×12], C22×C10 [×2], C22×C10 [×2], C4×Dic3, C2×D12, C2×C3⋊D4 [×4], C6×D4, C5×Dic3 [×4], C60 [×2], S3×C10 [×6], C2×C30, C2×C30 [×6], C4×C20, D4×C10, D4×C10 [×5], C123D4, C5×D12 [×2], C10×Dic3 [×2], C5×C3⋊D4 [×8], C2×C60, D4×C15 [×2], S3×C2×C10 [×2], C22×C30 [×2], C5×C41D4, Dic3×C20, C10×D12, C10×C3⋊D4 [×4], D4×C30, C5×C123D4
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×6], C23, C10 [×7], D6 [×3], C2×D4 [×3], C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C41D4, C5×D4 [×6], C22×C10, S3×D4 [×2], C2×C3⋊D4, S3×C10 [×3], D4×C10 [×3], C123D4, C5×C3⋊D4 [×2], S3×C2×C10, C5×C41D4, C5×S3×D4 [×2], C10×C3⋊D4, C5×C123D4

Smallest permutation representation of C5×C123D4
On 240 points
Generators in S240
(1 133 120 33 231)(2 134 109 34 232)(3 135 110 35 233)(4 136 111 36 234)(5 137 112 25 235)(6 138 113 26 236)(7 139 114 27 237)(8 140 115 28 238)(9 141 116 29 239)(10 142 117 30 240)(11 143 118 31 229)(12 144 119 32 230)(13 152 188 132 52)(14 153 189 121 53)(15 154 190 122 54)(16 155 191 123 55)(17 156 192 124 56)(18 145 181 125 57)(19 146 182 126 58)(20 147 183 127 59)(21 148 184 128 60)(22 149 185 129 49)(23 150 186 130 50)(24 151 187 131 51)(37 219 94 207 67)(38 220 95 208 68)(39 221 96 209 69)(40 222 85 210 70)(41 223 86 211 71)(42 224 87 212 72)(43 225 88 213 61)(44 226 89 214 62)(45 227 90 215 63)(46 228 91 216 64)(47 217 92 205 65)(48 218 93 206 66)(73 161 174 200 103)(74 162 175 201 104)(75 163 176 202 105)(76 164 177 203 106)(77 165 178 204 107)(78 166 179 193 108)(79 167 180 194 97)(80 168 169 195 98)(81 157 170 196 99)(82 158 171 197 100)(83 159 172 198 101)(84 160 173 199 102)
(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 41 77 128)(2 46 78 121)(3 39 79 126)(4 44 80 131)(5 37 81 124)(6 42 82 129)(7 47 83 122)(8 40 84 127)(9 45 73 132)(10 38 74 125)(11 43 75 130)(12 48 76 123)(13 116 90 174)(14 109 91 179)(15 114 92 172)(16 119 93 177)(17 112 94 170)(18 117 95 175)(19 110 96 180)(20 115 85 173)(21 120 86 178)(22 113 87 171)(23 118 88 176)(24 111 89 169)(25 207 196 156)(26 212 197 149)(27 205 198 154)(28 210 199 147)(29 215 200 152)(30 208 201 145)(31 213 202 150)(32 206 203 155)(33 211 204 148)(34 216 193 153)(35 209 194 146)(36 214 195 151)(49 138 224 158)(50 143 225 163)(51 136 226 168)(52 141 227 161)(53 134 228 166)(54 139 217 159)(55 144 218 164)(56 137 219 157)(57 142 220 162)(58 135 221 167)(59 140 222 160)(60 133 223 165)(61 105 186 229)(62 98 187 234)(63 103 188 239)(64 108 189 232)(65 101 190 237)(66 106 191 230)(67 99 192 235)(68 104 181 240)(69 97 182 233)(70 102 183 238)(71 107 184 231)(72 100 185 236)
(1 4)(2 3)(5 12)(6 11)(7 10)(8 9)(13 85)(14 96)(15 95)(16 94)(17 93)(18 92)(19 91)(20 90)(21 89)(22 88)(23 87)(24 86)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 123)(38 122)(39 121)(40 132)(41 131)(42 130)(43 129)(44 128)(45 127)(46 126)(47 125)(48 124)(49 225)(50 224)(51 223)(52 222)(53 221)(54 220)(55 219)(56 218)(57 217)(58 228)(59 227)(60 226)(61 185)(62 184)(63 183)(64 182)(65 181)(66 192)(67 191)(68 190)(69 189)(70 188)(71 187)(72 186)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 110)(111 120)(112 119)(113 118)(114 117)(115 116)(133 136)(134 135)(137 144)(138 143)(139 142)(140 141)(145 205)(146 216)(147 215)(148 214)(149 213)(150 212)(151 211)(152 210)(153 209)(154 208)(155 207)(156 206)(157 164)(158 163)(159 162)(160 161)(165 168)(166 167)(169 178)(170 177)(171 176)(172 175)(173 174)(179 180)(193 194)(195 204)(196 203)(197 202)(198 201)(199 200)(229 236)(230 235)(231 234)(232 233)(237 240)(238 239)

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B5C5D6A6B6C6D6E6F6G10A···10L10M···10T10U···10AB12A12B15A15B15C15D20A···20H20I···20X30A···30L30M···30AB60A···60H
order1222222234444445555666666610···1010···1010···1012121515151520···2020···2030···3030···3060···60
size11114412122226666111122244441···14···412···124422222···26···62···24···44···4

120 irreducible representations

dim111111111122222222222244
type+++++++++++
imageC1C2C2C2C2C5C10C10C10C10S3D4D4D6D6C3⋊D4C5×S3C5×D4C5×D4S3×C10S3×C10C5×C3⋊D4S3×D4C5×S3×D4
kernelC5×C123D4Dic3×C20C10×D12C10×C3⋊D4D4×C30C123D4C4×Dic3C2×D12C2×C3⋊D4C6×D4D4×C10C5×Dic3C60C2×C20C22×C10C20C2×D4Dic3C12C2×C4C23C4C10C2
# reps111414441641421244168481628

Matrix representation of C5×C123D4 in GL4(𝔽61) generated by

1000
0100
00580
00058
,
0100
60100
00159
00160
,
43900
521800
0010
0001
,
06000
60000
00159
00060
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,58,0,0,0,0,58],[0,60,0,0,1,1,0,0,0,0,1,1,0,0,59,60],[43,52,0,0,9,18,0,0,0,0,1,0,0,0,0,1],[0,60,0,0,60,0,0,0,0,0,1,0,0,0,59,60] >;

C5×C123D4 in GAP, Magma, Sage, TeX

C_5\times C_{12}\rtimes_3D_4
% in TeX

G:=Group("C5xC12:3D4");
// GroupNames label

G:=SmallGroup(480,819);
// by ID

G=gap.SmallGroup(480,819);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,568,926,891,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽