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G = C5×C12.48D4order 480 = 25·3·5

Direct product of C5 and C12.48D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C12.48D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — C10×Dic6 — C5×C12.48D4
 Lower central C3 — C2×C6 — C5×C12.48D4
 Upper central C1 — C2×C10 — C22×C20

Generators and relations for C5×C12.48D4
G = < a,b,c,d | a5=b12=c4=1, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b6c-1 >

Subgroups: 292 in 148 conjugacy classes, 74 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C5, C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C10 [×3], C10 [×2], Dic3 [×4], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C20 [×2], C20 [×5], C2×C10, C2×C10 [×2], C2×C10 [×2], Dic6 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×2], C22⋊Q8, C2×C20 [×2], C2×C20 [×6], C5×Q8 [×2], C22×C10, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C2×Dic6, C22×C12, C5×Dic3 [×4], C60 [×2], C60, C2×C30, C2×C30 [×2], C2×C30 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×3], C22×C20, Q8×C10, C12.48D4, C5×Dic6 [×2], C10×Dic3 [×4], C2×C60 [×2], C2×C60 [×2], C22×C30, C5×C22⋊Q8, C5×Dic3⋊C4 [×2], C5×C4⋊Dic3, C5×C6.D4 [×2], C10×Dic6, C22×C60, C5×C12.48D4
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], Q8 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×Q8, C4○D4, C2×C10 [×7], Dic6 [×2], C3⋊D4 [×2], C22×S3, C5×S3, C22⋊Q8, C5×D4 [×2], C5×Q8 [×2], C22×C10, C2×Dic6, C4○D12, C2×C3⋊D4, S3×C10 [×3], D4×C10, Q8×C10, C5×C4○D4, C12.48D4, C5×Dic6 [×2], C5×C3⋊D4 [×2], S3×C2×C10, C5×C22⋊Q8, C10×Dic6, C5×C4○D12, C10×C3⋊D4, C5×C12.48D4

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

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A ··· 6G 10A ··· 10L 10M ··· 10T 12A ··· 12H 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AF 30A ··· 30AB 60A ··· 60AF order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 5 5 6 ··· 6 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 2 2 2 2 2 12 12 12 12 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 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 Q8 D6 D6 C4○D4 C3⋊D4 Dic6 C5×S3 C5×D4 C5×Q8 C4○D12 S3×C10 S3×C10 C5×C4○D4 C5×C3⋊D4 C5×Dic6 C5×C4○D12 kernel C5×C12.48D4 C5×Dic3⋊C4 C5×C4⋊Dic3 C5×C6.D4 C10×Dic6 C22×C60 C12.48D4 Dic3⋊C4 C4⋊Dic3 C6.D4 C2×Dic6 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 2 1 2 1 1 4 8 4 8 4 4 1 2 2 2 1 2 4 4 4 8 8 4 8 4 8 16 16 16

Matrix representation of C5×C12.48D4 in GL4(𝔽61) generated by

 34 0 0 0 0 34 0 0 0 0 34 0 0 0 0 34
,
 1 0 0 0 0 1 0 0 0 0 32 0 0 0 0 21
,
 1 51 0 0 49 60 0 0 0 0 0 1 0 0 60 0
,
 1 51 0 0 0 60 0 0 0 0 0 1 0 0 60 0
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,21],[1,49,0,0,51,60,0,0,0,0,0,60,0,0,1,0],[1,0,0,0,51,60,0,0,0,0,0,60,0,0,1,0] >;

C5×C12.48D4 in GAP, Magma, Sage, TeX

C_5\times C_{12}._{48}D_4
% in TeX

G:=Group("C5xC12.48D4");
// GroupNames label

G:=SmallGroup(480,803);
// by ID

G=gap.SmallGroup(480,803);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,1149,568,926,15686]);
// Polycyclic

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

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