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G = C5×C12.55D4order 480 = 25·3·5

Direct product of C5 and C12.55D4

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

Aliases: C5×C12.55D4, C60.240D4, C30.42M4(2), (C2×C6)⋊2C40, (C2×C30)⋊10C8, C30.72(C2×C8), (C2×C60).31C4, C6.10(C2×C40), (C2×C12).5C20, C12.55(C5×D4), C1515(C22⋊C8), (C2×C20).449D6, (C22×C6).5C20, (C22×C20).5S3, C6.6(C5×M4(2)), (C22×C60).30C2, (C22×C30).21C4, (C2×C20).15Dic3, C20.123(C3⋊D4), C23.3(C5×Dic3), (C22×C12).11C10, (C2×C60).561C22, C22.9(C10×Dic3), C30.116(C22⋊C4), C10.14(C4.Dic3), (C22×C10).10Dic3, C10.31(C6.D4), C2.5(C10×C3⋊C8), C32(C5×C22⋊C8), C222(C5×C3⋊C8), (C10×C3⋊C8)⋊24C2, (C2×C3⋊C8)⋊10C10, (C2×C10)⋊6(C3⋊C8), C10.24(C2×C3⋊C8), C4.30(C5×C3⋊D4), (C2×C4).96(S3×C10), (C2×C6).27(C2×C20), C6.11(C5×C22⋊C4), (C22×C4).3(C5×S3), (C2×C4).3(C5×Dic3), (C2×C30).195(C2×C4), C2.3(C5×C4.Dic3), C2.1(C5×C6.D4), (C2×C12).113(C2×C10), (C2×C10).61(C2×Dic3), SmallGroup(480,149)

Series: Derived Chief Lower central Upper central

C1C6 — C5×C12.55D4
C1C3C6C2×C6C2×C12C2×C60C10×C3⋊C8 — C5×C12.55D4
C3C6 — C5×C12.55D4
C1C2×C20C22×C20

Generators and relations for C5×C12.55D4
 G = < a,b,c,d | a5=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b3c3 >

Subgroups: 164 in 100 conjugacy classes, 58 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×2], C22 [×2], C5, C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×2], C23, C10 [×3], C10 [×2], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C2×C8 [×2], C22×C4, C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C3⋊C8 [×2], C2×C12 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×2], C22⋊C8, C40 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C2×C3⋊C8 [×2], C22×C12, C60 [×2], C60, C2×C30, C2×C30 [×2], C2×C30 [×2], C2×C40 [×2], C22×C20, C12.55D4, C5×C3⋊C8 [×2], C2×C60 [×2], C2×C60 [×2], C22×C30, C5×C22⋊C8, C10×C3⋊C8 [×2], C22×C60, C5×C12.55D4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C8 [×2], C2×C4, D4 [×2], C10 [×3], Dic3 [×2], D6, C22⋊C4, C2×C8, M4(2), C20 [×2], C2×C10, C3⋊C8 [×2], C2×Dic3, C3⋊D4 [×2], C5×S3, C22⋊C8, C40 [×2], C2×C20, C5×D4 [×2], C2×C3⋊C8, C4.Dic3, C6.D4, C5×Dic3 [×2], S3×C10, C5×C22⋊C4, C2×C40, C5×M4(2), C12.55D4, C5×C3⋊C8 [×2], C10×Dic3, C5×C3⋊D4 [×2], C5×C22⋊C8, C10×C3⋊C8, C5×C4.Dic3, C5×C6.D4, C5×C12.55D4

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

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

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

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

180 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F5A5B5C5D6A···6G8A···8H10A···10L10M···10T12A···12H15A15B15C15D20A···20P20Q···20X30A···30AB40A···40AF60A···60AF
order122222344444455556···68···810···1010···1012···121515151520···2020···2030···3040···4060···60
size111122211112211112···26···61···12···22···222221···12···22···26···62···2

180 irreducible representations

dim111111111111222222222222222222
type+++++-+-
imageC1C2C2C4C4C5C8C10C10C20C20C40S3D4Dic3D6Dic3M4(2)C3⋊D4C3⋊C8C5×S3C5×D4C4.Dic3C5×Dic3S3×C10C5×Dic3C5×M4(2)C5×C3⋊D4C5×C3⋊C8C5×C4.Dic3
kernelC5×C12.55D4C10×C3⋊C8C22×C60C2×C60C22×C30C12.55D4C2×C30C2×C3⋊C8C22×C12C2×C12C22×C6C2×C6C22×C20C60C2×C20C2×C20C22×C10C30C20C2×C10C22×C4C12C10C2×C4C2×C4C23C6C4C22C2
# reps1212248848832121112444844448161616

Matrix representation of C5×C12.55D4 in GL4(𝔽241) generated by

1000
0100
00910
00091
,
240000
024000
002370
000181
,
0100
1000
0001
001770
,
0100
240000
0001
00640
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,91,0,0,0,0,91],[240,0,0,0,0,240,0,0,0,0,237,0,0,0,0,181],[0,1,0,0,1,0,0,0,0,0,0,177,0,0,1,0],[0,240,0,0,1,0,0,0,0,0,0,64,0,0,1,0] >;

C5×C12.55D4 in GAP, Magma, Sage, TeX

C_5\times C_{12}._{55}D_4
% in TeX

G:=Group("C5xC12.55D4");
// GroupNames label

G:=SmallGroup(480,149);
// by ID

G=gap.SmallGroup(480,149);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,136,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
// generators/relations

׿
×
𝔽