direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12⋊7D4, C60⋊32D4, C12⋊7(C5×D4), D6⋊C4⋊3C10, (C2×C10)⋊7D12, (C2×C30)⋊27D4, (C2×D12)⋊6C10, C4⋊Dic3⋊9C10, C6.43(D4×C10), C22⋊2(C5×D12), (C10×D12)⋊22C2, C20⋊15(C3⋊D4), C15⋊33(C4⋊D4), (C22×C12)⋊6C10, (C22×C20)⋊15S3, (C22×C60)⋊18C2, C2.17(C10×D12), (C2×C20).440D6, C30.426(C2×D4), C10.88(C2×D12), C23.29(S3×C10), C30.212(C4○D4), (C2×C30).427C23, (C2×C60).532C22, (C22×C10).126D6, C10.126(C4○D12), (C22×C30).178C22, (C10×Dic3).149C22, (C2×C6)⋊5(C5×D4), C4⋊3(C5×C3⋊D4), C3⋊3(C5×C4⋊D4), (C5×D6⋊C4)⋊3C2, (C2×C3⋊D4)⋊3C10, (C22×C4)⋊6(C5×S3), C6.17(C5×C4○D4), C2.7(C10×C3⋊D4), (C10×C3⋊D4)⋊18C2, (C2×C4).87(S3×C10), (C5×C4⋊Dic3)⋊27C2, C2.19(C5×C4○D12), C22.56(S3×C2×C10), (C2×C12).99(C2×C10), C10.128(C2×C3⋊D4), (S3×C2×C10).71C22, (C2×C6).48(C22×C10), (C22×C6).40(C2×C10), (C22×S3).10(C2×C10), (C2×C10).361(C22×S3), (C2×Dic3).13(C2×C10), SmallGroup(480,809)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C12⋊7D4
G = < a,b,c,d | a5=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 484 in 188 conjugacy classes, 74 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C4⋊D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C4⋊Dic3, D6⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C5×Dic3, C60, C60, S3×C10, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C12⋊7D4, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, C2×C60, S3×C2×C10, C22×C30, C5×C4⋊D4, C5×C4⋊Dic3, C5×D6⋊C4, C10×D12, C10×C3⋊D4, C22×C60, C5×C12⋊7D4
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C4○D4, C2×C10, D12, C3⋊D4, C22×S3, C5×S3, C4⋊D4, C5×D4, C22×C10, C2×D12, C4○D12, C2×C3⋊D4, S3×C10, D4×C10, C5×C4○D4, C12⋊7D4, C5×D12, C5×C3⋊D4, S3×C2×C10, C5×C4⋊D4, C10×D12, C5×C4○D12, C10×C3⋊D4, C5×C12⋊7D4
►On 240 points
Generators in S240
(1 136 58 150 79)(2 137 59 151 80)(3 138 60 152 81)(4 139 49 153 82)(5 140 50 154 83)(6 141 51 155 84)(7 142 52 156 73)(8 143 53 145 74)(9 144 54 146 75)(10 133 55 147 76)(11 134 56 148 77)(12 135 57 149 78)(13 229 30 202 183)(14 230 31 203 184)(15 231 32 204 185)(16 232 33 193 186)(17 233 34 194 187)(18 234 35 195 188)(19 235 36 196 189)(20 236 25 197 190)(21 237 26 198 191)(22 238 27 199 192)(23 239 28 200 181)(24 240 29 201 182)(37 124 94 162 107)(38 125 95 163 108)(39 126 96 164 97)(40 127 85 165 98)(41 128 86 166 99)(42 129 87 167 100)(43 130 88 168 101)(44 131 89 157 102)(45 132 90 158 103)(46 121 91 159 104)(47 122 92 160 105)(48 123 93 161 106)(61 212 217 116 173)(62 213 218 117 174)(63 214 219 118 175)(64 215 220 119 176)(65 216 221 120 177)(66 205 222 109 178)(67 206 223 110 179)(68 207 224 111 180)(69 208 225 112 169)(70 209 226 113 170)(71 210 227 114 171)(72 211 228 115 172)
(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 41 223 30)(2 40 224 29)(3 39 225 28)(4 38 226 27)(5 37 227 26)(6 48 228 25)(7 47 217 36)(8 46 218 35)(9 45 219 34)(10 44 220 33)(11 43 221 32)(12 42 222 31)(13 150 166 67)(14 149 167 66)(15 148 168 65)(16 147 157 64)(17 146 158 63)(18 145 159 62)(19 156 160 61)(20 155 161 72)(21 154 162 71)(22 153 163 70)(23 152 164 69)(24 151 165 68)(49 95 170 192)(50 94 171 191)(51 93 172 190)(52 92 173 189)(53 91 174 188)(54 90 175 187)(55 89 176 186)(56 88 177 185)(57 87 178 184)(58 86 179 183)(59 85 180 182)(60 96 169 181)(73 105 212 235)(74 104 213 234)(75 103 214 233)(76 102 215 232)(77 101 216 231)(78 100 205 230)(79 99 206 229)(80 98 207 240)(81 97 208 239)(82 108 209 238)(83 107 210 237)(84 106 211 236)(109 203 135 129)(110 202 136 128)(111 201 137 127)(112 200 138 126)(113 199 139 125)(114 198 140 124)(115 197 141 123)(116 196 142 122)(117 195 143 121)(118 194 144 132)(119 193 133 131)(120 204 134 130)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 150)(14 149)(15 148)(16 147)(17 146)(18 145)(19 156)(20 155)(21 154)(22 153)(23 152)(24 151)(37 227)(38 226)(39 225)(40 224)(41 223)(42 222)(43 221)(44 220)(45 219)(46 218)(47 217)(48 228)(49 192)(50 191)(51 190)(52 189)(53 188)(54 187)(55 186)(56 185)(57 184)(58 183)(59 182)(60 181)(61 160)(62 159)(63 158)(64 157)(65 168)(66 167)(67 166)(68 165)(69 164)(70 163)(71 162)(72 161)(73 235)(74 234)(75 233)(76 232)(77 231)(78 230)(79 229)(80 240)(81 239)(82 238)(83 237)(84 236)(85 180)(86 179)(87 178)(88 177)(89 176)(90 175)(91 174)(92 173)(93 172)(94 171)(95 170)(96 169)(97 208)(98 207)(99 206)(100 205)(101 216)(102 215)(103 214)(104 213)(105 212)(106 211)(107 210)(108 209)(109 129)(110 128)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 132)(119 131)(120 130)(133 193)(134 204)(135 203)(136 202)(137 201)(138 200)(139 199)(140 198)(141 197)(142 196)(143 195)(144 194)
G:=sub<Sym(240)| (1,136,58,150,79)(2,137,59,151,80)(3,138,60,152,81)(4,139,49,153,82)(5,140,50,154,83)(6,141,51,155,84)(7,142,52,156,73)(8,143,53,145,74)(9,144,54,146,75)(10,133,55,147,76)(11,134,56,148,77)(12,135,57,149,78)(13,229,30,202,183)(14,230,31,203,184)(15,231,32,204,185)(16,232,33,193,186)(17,233,34,194,187)(18,234,35,195,188)(19,235,36,196,189)(20,236,25,197,190)(21,237,26,198,191)(22,238,27,199,192)(23,239,28,200,181)(24,240,29,201,182)(37,124,94,162,107)(38,125,95,163,108)(39,126,96,164,97)(40,127,85,165,98)(41,128,86,166,99)(42,129,87,167,100)(43,130,88,168,101)(44,131,89,157,102)(45,132,90,158,103)(46,121,91,159,104)(47,122,92,160,105)(48,123,93,161,106)(61,212,217,116,173)(62,213,218,117,174)(63,214,219,118,175)(64,215,220,119,176)(65,216,221,120,177)(66,205,222,109,178)(67,206,223,110,179)(68,207,224,111,180)(69,208,225,112,169)(70,209,226,113,170)(71,210,227,114,171)(72,211,228,115,172), (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), (1,41,223,30)(2,40,224,29)(3,39,225,28)(4,38,226,27)(5,37,227,26)(6,48,228,25)(7,47,217,36)(8,46,218,35)(9,45,219,34)(10,44,220,33)(11,43,221,32)(12,42,222,31)(13,150,166,67)(14,149,167,66)(15,148,168,65)(16,147,157,64)(17,146,158,63)(18,145,159,62)(19,156,160,61)(20,155,161,72)(21,154,162,71)(22,153,163,70)(23,152,164,69)(24,151,165,68)(49,95,170,192)(50,94,171,191)(51,93,172,190)(52,92,173,189)(53,91,174,188)(54,90,175,187)(55,89,176,186)(56,88,177,185)(57,87,178,184)(58,86,179,183)(59,85,180,182)(60,96,169,181)(73,105,212,235)(74,104,213,234)(75,103,214,233)(76,102,215,232)(77,101,216,231)(78,100,205,230)(79,99,206,229)(80,98,207,240)(81,97,208,239)(82,108,209,238)(83,107,210,237)(84,106,211,236)(109,203,135,129)(110,202,136,128)(111,201,137,127)(112,200,138,126)(113,199,139,125)(114,198,140,124)(115,197,141,123)(116,196,142,122)(117,195,143,121)(118,194,144,132)(119,193,133,131)(120,204,134,130), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,150)(14,149)(15,148)(16,147)(17,146)(18,145)(19,156)(20,155)(21,154)(22,153)(23,152)(24,151)(37,227)(38,226)(39,225)(40,224)(41,223)(42,222)(43,221)(44,220)(45,219)(46,218)(47,217)(48,228)(49,192)(50,191)(51,190)(52,189)(53,188)(54,187)(55,186)(56,185)(57,184)(58,183)(59,182)(60,181)(61,160)(62,159)(63,158)(64,157)(65,168)(66,167)(67,166)(68,165)(69,164)(70,163)(71,162)(72,161)(73,235)(74,234)(75,233)(76,232)(77,231)(78,230)(79,229)(80,240)(81,239)(82,238)(83,237)(84,236)(85,180)(86,179)(87,178)(88,177)(89,176)(90,175)(91,174)(92,173)(93,172)(94,171)(95,170)(96,169)(97,208)(98,207)(99,206)(100,205)(101,216)(102,215)(103,214)(104,213)(105,212)(106,211)(107,210)(108,209)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,132)(119,131)(120,130)(133,193)(134,204)(135,203)(136,202)(137,201)(138,200)(139,199)(140,198)(141,197)(142,196)(143,195)(144,194)>;
G:=Group( (1,136,58,150,79)(2,137,59,151,80)(3,138,60,152,81)(4,139,49,153,82)(5,140,50,154,83)(6,141,51,155,84)(7,142,52,156,73)(8,143,53,145,74)(9,144,54,146,75)(10,133,55,147,76)(11,134,56,148,77)(12,135,57,149,78)(13,229,30,202,183)(14,230,31,203,184)(15,231,32,204,185)(16,232,33,193,186)(17,233,34,194,187)(18,234,35,195,188)(19,235,36,196,189)(20,236,25,197,190)(21,237,26,198,191)(22,238,27,199,192)(23,239,28,200,181)(24,240,29,201,182)(37,124,94,162,107)(38,125,95,163,108)(39,126,96,164,97)(40,127,85,165,98)(41,128,86,166,99)(42,129,87,167,100)(43,130,88,168,101)(44,131,89,157,102)(45,132,90,158,103)(46,121,91,159,104)(47,122,92,160,105)(48,123,93,161,106)(61,212,217,116,173)(62,213,218,117,174)(63,214,219,118,175)(64,215,220,119,176)(65,216,221,120,177)(66,205,222,109,178)(67,206,223,110,179)(68,207,224,111,180)(69,208,225,112,169)(70,209,226,113,170)(71,210,227,114,171)(72,211,228,115,172), (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), (1,41,223,30)(2,40,224,29)(3,39,225,28)(4,38,226,27)(5,37,227,26)(6,48,228,25)(7,47,217,36)(8,46,218,35)(9,45,219,34)(10,44,220,33)(11,43,221,32)(12,42,222,31)(13,150,166,67)(14,149,167,66)(15,148,168,65)(16,147,157,64)(17,146,158,63)(18,145,159,62)(19,156,160,61)(20,155,161,72)(21,154,162,71)(22,153,163,70)(23,152,164,69)(24,151,165,68)(49,95,170,192)(50,94,171,191)(51,93,172,190)(52,92,173,189)(53,91,174,188)(54,90,175,187)(55,89,176,186)(56,88,177,185)(57,87,178,184)(58,86,179,183)(59,85,180,182)(60,96,169,181)(73,105,212,235)(74,104,213,234)(75,103,214,233)(76,102,215,232)(77,101,216,231)(78,100,205,230)(79,99,206,229)(80,98,207,240)(81,97,208,239)(82,108,209,238)(83,107,210,237)(84,106,211,236)(109,203,135,129)(110,202,136,128)(111,201,137,127)(112,200,138,126)(113,199,139,125)(114,198,140,124)(115,197,141,123)(116,196,142,122)(117,195,143,121)(118,194,144,132)(119,193,133,131)(120,204,134,130), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,150)(14,149)(15,148)(16,147)(17,146)(18,145)(19,156)(20,155)(21,154)(22,153)(23,152)(24,151)(37,227)(38,226)(39,225)(40,224)(41,223)(42,222)(43,221)(44,220)(45,219)(46,218)(47,217)(48,228)(49,192)(50,191)(51,190)(52,189)(53,188)(54,187)(55,186)(56,185)(57,184)(58,183)(59,182)(60,181)(61,160)(62,159)(63,158)(64,157)(65,168)(66,167)(67,166)(68,165)(69,164)(70,163)(71,162)(72,161)(73,235)(74,234)(75,233)(76,232)(77,231)(78,230)(79,229)(80,240)(81,239)(82,238)(83,237)(84,236)(85,180)(86,179)(87,178)(88,177)(89,176)(90,175)(91,174)(92,173)(93,172)(94,171)(95,170)(96,169)(97,208)(98,207)(99,206)(100,205)(101,216)(102,215)(103,214)(104,213)(105,212)(106,211)(107,210)(108,209)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,132)(119,131)(120,130)(133,193)(134,204)(135,203)(136,202)(137,201)(138,200)(139,199)(140,198)(141,197)(142,196)(143,195)(144,194) );
G=PermutationGroup([[(1,136,58,150,79),(2,137,59,151,80),(3,138,60,152,81),(4,139,49,153,82),(5,140,50,154,83),(6,141,51,155,84),(7,142,52,156,73),(8,143,53,145,74),(9,144,54,146,75),(10,133,55,147,76),(11,134,56,148,77),(12,135,57,149,78),(13,229,30,202,183),(14,230,31,203,184),(15,231,32,204,185),(16,232,33,193,186),(17,233,34,194,187),(18,234,35,195,188),(19,235,36,196,189),(20,236,25,197,190),(21,237,26,198,191),(22,238,27,199,192),(23,239,28,200,181),(24,240,29,201,182),(37,124,94,162,107),(38,125,95,163,108),(39,126,96,164,97),(40,127,85,165,98),(41,128,86,166,99),(42,129,87,167,100),(43,130,88,168,101),(44,131,89,157,102),(45,132,90,158,103),(46,121,91,159,104),(47,122,92,160,105),(48,123,93,161,106),(61,212,217,116,173),(62,213,218,117,174),(63,214,219,118,175),(64,215,220,119,176),(65,216,221,120,177),(66,205,222,109,178),(67,206,223,110,179),(68,207,224,111,180),(69,208,225,112,169),(70,209,226,113,170),(71,210,227,114,171),(72,211,228,115,172)], [(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)], [(1,41,223,30),(2,40,224,29),(3,39,225,28),(4,38,226,27),(5,37,227,26),(6,48,228,25),(7,47,217,36),(8,46,218,35),(9,45,219,34),(10,44,220,33),(11,43,221,32),(12,42,222,31),(13,150,166,67),(14,149,167,66),(15,148,168,65),(16,147,157,64),(17,146,158,63),(18,145,159,62),(19,156,160,61),(20,155,161,72),(21,154,162,71),(22,153,163,70),(23,152,164,69),(24,151,165,68),(49,95,170,192),(50,94,171,191),(51,93,172,190),(52,92,173,189),(53,91,174,188),(54,90,175,187),(55,89,176,186),(56,88,177,185),(57,87,178,184),(58,86,179,183),(59,85,180,182),(60,96,169,181),(73,105,212,235),(74,104,213,234),(75,103,214,233),(76,102,215,232),(77,101,216,231),(78,100,205,230),(79,99,206,229),(80,98,207,240),(81,97,208,239),(82,108,209,238),(83,107,210,237),(84,106,211,236),(109,203,135,129),(110,202,136,128),(111,201,137,127),(112,200,138,126),(113,199,139,125),(114,198,140,124),(115,197,141,123),(116,196,142,122),(117,195,143,121),(118,194,144,132),(119,193,133,131),(120,204,134,130)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,150),(14,149),(15,148),(16,147),(17,146),(18,145),(19,156),(20,155),(21,154),(22,153),(23,152),(24,151),(37,227),(38,226),(39,225),(40,224),(41,223),(42,222),(43,221),(44,220),(45,219),(46,218),(47,217),(48,228),(49,192),(50,191),(51,190),(52,189),(53,188),(54,187),(55,186),(56,185),(57,184),(58,183),(59,182),(60,181),(61,160),(62,159),(63,158),(64,157),(65,168),(66,167),(67,166),(68,165),(69,164),(70,163),(71,162),(72,161),(73,235),(74,234),(75,233),(76,232),(77,231),(78,230),(79,229),(80,240),(81,239),(82,238),(83,237),(84,236),(85,180),(86,179),(87,178),(88,177),(89,176),(90,175),(91,174),(92,173),(93,172),(94,171),(95,170),(96,169),(97,208),(98,207),(99,206),(100,205),(101,216),(102,215),(103,214),(104,213),(105,212),(106,211),(107,210),(108,209),(109,129),(110,128),(111,127),(112,126),(113,125),(114,124),(115,123),(116,122),(117,121),(118,132),(119,131),(120,130),(133,193),(134,204),(135,203),(136,202),(137,201),(138,200),(139,199),(140,198),(141,197),(142,196),(143,195),(144,194)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20X | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | C5×S3 | C5×D4 | C5×D4 | C4○D12 | S3×C10 | S3×C10 | C5×C4○D4 | C5×C3⋊D4 | C5×D12 | C5×C4○D12 |
| kernel | C5×C12⋊7D4 | C5×C4⋊Dic3 | C5×D6⋊C4 | C10×D12 | C10×C3⋊D4 | C22×C60 | C12⋊7D4 | C4⋊Dic3 | D6⋊C4 | C2×D12 | C2×C3⋊D4 | C22×C12 | C22×C20 | C60 | C2×C30 | C2×C20 | C22×C10 | C30 | C20 | C2×C10 | C22×C4 | C12 | C2×C6 | C10 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 8 | 4 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 8 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C5×C12⋊7D4 ►in GL4(𝔽61) generated by
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 34 |
| 32 | 0 | 0 | 0 |
| 0 | 21 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[32,0,0,0,0,21,0,0,0,0,60,0,0,0,0,60],[0,1,0,0,1,0,0,0,0,0,0,60,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C5×C12⋊7D4 in GAP, Magma, Sage, TeX
C_5\times C_{12}\rtimes_7D_4 % in TeX
G:=Group("C5xC12:7D4"); // GroupNames label
G:=SmallGroup(480,809);
// by ID
G=gap.SmallGroup(480,809);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,568,926,15686]);
// 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=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations