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G = C5×C12.23D4order 480 = 25·3·5

Direct product of C5 and C12.23D4

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

Aliases: C5×C12.23D4, C60.156D4, (C6×Q8)⋊4C10, D6⋊C416C10, (Q8×C10)⋊15S3, (Q8×C30)⋊18C2, C12.23(C5×D4), C6.58(D4×C10), (C4×Dic3)⋊7C10, (C2×D12).9C10, C30.441(C2×D4), (C2×C20).247D6, (Dic3×C20)⋊19C2, (C10×D12).19C2, C20.76(C3⋊D4), C1527(C4.4D4), C30.273(C4○D4), (C2×C30).438C23, (C2×C60).368C22, C10.55(Q83S3), (C10×Dic3).233C22, (C2×Q8)⋊6(C5×S3), C34(C5×C4.4D4), (C5×D6⋊C4)⋊38C2, C6.37(C5×C4○D4), C4.11(C5×C3⋊D4), (C2×C4).57(S3×C10), C2.22(C10×C3⋊D4), C22.65(S3×C2×C10), C2.9(C5×Q83S3), (C2×C12).42(C2×C10), C10.143(C2×C3⋊D4), (S3×C2×C10).74C22, (C2×C6).59(C22×C10), (C22×S3).13(C2×C10), (C2×C10).372(C22×S3), (C2×Dic3).42(C2×C10), SmallGroup(480,826)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×C12.23D4
C1C3C6C2×C6C2×C30S3×C2×C10C10×D12 — C5×C12.23D4
C3C2×C6 — C5×C12.23D4
C1C2×C10Q8×C10

Generators and relations for C5×C12.23D4
 G = < a,b,c,d | a5=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=b6c-1 >

Subgroups: 388 in 152 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3 [×2], C5×S3 [×2], C30, C30 [×2], C4.4D4, C2×C20, C2×C20 [×2], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×C10 [×2], C4×Dic3, D6⋊C4 [×4], C2×D12, C6×Q8, C5×Dic3 [×2], C60 [×2], C60 [×2], S3×C10 [×6], C2×C30, C4×C20, C5×C22⋊C4 [×4], D4×C10, Q8×C10, C12.23D4, C5×D12 [×2], C10×Dic3 [×2], C2×C60, C2×C60 [×2], Q8×C15 [×2], S3×C2×C10 [×2], C5×C4.4D4, Dic3×C20, C5×D6⋊C4 [×4], C10×D12, Q8×C30, C5×C12.23D4
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C4○D4 [×2], C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C4.4D4, C5×D4 [×2], C22×C10, Q83S3 [×2], C2×C3⋊D4, S3×C10 [×3], D4×C10, C5×C4○D4 [×2], C12.23D4, C5×C3⋊D4 [×2], S3×C2×C10, C5×C4.4D4, C5×Q83S3 [×2], C10×C3⋊D4, C5×C12.23D4

Smallest permutation representation of C5×C12.23D4
On 240 points
Generators in S240
(1 211 95 199 83)(2 212 96 200 84)(3 213 85 201 73)(4 214 86 202 74)(5 215 87 203 75)(6 216 88 204 76)(7 205 89 193 77)(8 206 90 194 78)(9 207 91 195 79)(10 208 92 196 80)(11 209 93 197 81)(12 210 94 198 82)(13 37 153 25 141)(14 38 154 26 142)(15 39 155 27 143)(16 40 156 28 144)(17 41 145 29 133)(18 42 146 30 134)(19 43 147 31 135)(20 44 148 32 136)(21 45 149 33 137)(22 46 150 34 138)(23 47 151 35 139)(24 48 152 36 140)(49 165 170 184 72)(50 166 171 185 61)(51 167 172 186 62)(52 168 173 187 63)(53 157 174 188 64)(54 158 175 189 65)(55 159 176 190 66)(56 160 177 191 67)(57 161 178 192 68)(58 162 179 181 69)(59 163 180 182 70)(60 164 169 183 71)(97 128 237 109 225)(98 129 238 110 226)(99 130 239 111 227)(100 131 240 112 228)(101 132 229 113 217)(102 121 230 114 218)(103 122 231 115 219)(104 123 232 116 220)(105 124 233 117 221)(106 125 234 118 222)(107 126 235 119 223)(108 127 236 120 224)
(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)
(1 174 23 104)(2 179 24 97)(3 172 13 102)(4 177 14 107)(5 170 15 100)(6 175 16 105)(7 180 17 98)(8 173 18 103)(9 178 19 108)(10 171 20 101)(11 176 21 106)(12 169 22 99)(25 114 201 51)(26 119 202 56)(27 112 203 49)(28 117 204 54)(29 110 193 59)(30 115 194 52)(31 120 195 57)(32 113 196 50)(33 118 197 55)(34 111 198 60)(35 116 199 53)(36 109 200 58)(37 121 213 186)(38 126 214 191)(39 131 215 184)(40 124 216 189)(41 129 205 182)(42 122 206 187)(43 127 207 192)(44 132 208 185)(45 125 209 190)(46 130 210 183)(47 123 211 188)(48 128 212 181)(61 148 229 92)(62 153 230 85)(63 146 231 90)(64 151 232 95)(65 156 233 88)(66 149 234 93)(67 154 235 86)(68 147 236 91)(69 152 237 96)(70 145 238 89)(71 150 239 94)(72 155 240 87)(73 167 141 218)(74 160 142 223)(75 165 143 228)(76 158 144 221)(77 163 133 226)(78 168 134 219)(79 161 135 224)(80 166 136 217)(81 159 137 222)(82 164 138 227)(83 157 139 220)(84 162 140 225)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 24)(10 23)(11 22)(12 21)(25 194)(26 193)(27 204)(28 203)(29 202)(30 201)(31 200)(32 199)(33 198)(34 197)(35 196)(36 195)(37 206)(38 205)(39 216)(40 215)(41 214)(42 213)(43 212)(44 211)(45 210)(46 209)(47 208)(48 207)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 70)(62 69)(63 68)(64 67)(65 66)(71 72)(73 134)(74 133)(75 144)(76 143)(77 142)(78 141)(79 140)(80 139)(81 138)(82 137)(83 136)(84 135)(85 146)(86 145)(87 156)(88 155)(89 154)(90 153)(91 152)(92 151)(93 150)(94 149)(95 148)(96 147)(97 102)(98 101)(99 100)(103 108)(104 107)(105 106)(109 114)(110 113)(111 112)(115 120)(116 119)(117 118)(121 128)(122 127)(123 126)(124 125)(129 132)(130 131)(157 160)(158 159)(161 168)(162 167)(163 166)(164 165)(169 170)(171 180)(172 179)(173 178)(174 177)(175 176)(181 186)(182 185)(183 184)(187 192)(188 191)(189 190)(217 226)(218 225)(219 224)(220 223)(221 222)(227 228)(229 238)(230 237)(231 236)(232 235)(233 234)(239 240)

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B5C5D6A6B6C10A···10L10M···10T12A···12F15A15B15C15D20A···20H20I···20P20Q···20AF30A···30L60A···60X
order122222344444444555566610···1010···1012···121515151520···2020···2020···2030···3060···60
size1111121222244666611112221···112···124···422222···24···46···62···24···4

120 irreducible representations

dim1111111111222222222244
type+++++++++
imageC1C2C2C2C2C5C10C10C10C10S3D4D6C4○D4C3⋊D4C5×S3C5×D4S3×C10C5×C4○D4C5×C3⋊D4Q83S3C5×Q83S3
kernelC5×C12.23D4Dic3×C20C5×D6⋊C4C10×D12Q8×C30C12.23D4C4×Dic3D6⋊C4C2×D12C6×Q8Q8×C10C60C2×C20C30C20C2×Q8C12C2×C4C6C4C10C2
# reps11411441644123444812161628

Matrix representation of C5×C12.23D4 in GL4(𝔽61) generated by

20000
02000
0010
0001
,
0100
60100
00011
00110
,
525200
43900
00500
00050
,
16000
06000
00011
00500
G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,1,0,0,0,0,1],[0,60,0,0,1,1,0,0,0,0,0,11,0,0,11,0],[52,43,0,0,52,9,0,0,0,0,50,0,0,0,0,50],[1,0,0,0,60,60,0,0,0,0,0,50,0,0,11,0] >;

C5×C12.23D4 in GAP, Magma, Sage, TeX

C_5\times C_{12}._{23}D_4
% in TeX

G:=Group("C5xC12.23D4");
// GroupNames label

G:=SmallGroup(480,826);
// by ID

G=gap.SmallGroup(480,826);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,568,926,891,436,15686]);
// 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=b^6*c^-1>;
// generators/relations

׿
×
𝔽