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## G = C5×C12⋊7D4order 480 = 25·3·5

### Direct product of C5 and C12⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C12⋊7D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — C10×D12 — C5×C12⋊7D4
 Lower central C3 — C2×C6 — C5×C12⋊7D4
 Upper central C1 — C2×C10 — C22×C20

Generators and relations for C5×C127D4
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 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C10 [×3], C10 [×4], Dic3 [×2], C12 [×2], C12, D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C20 [×2], C20 [×3], C2×C10, C2×C10 [×2], C2×C10 [×8], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C22×S3 [×2], C22×C6, C5×S3 [×2], C30 [×3], C30 [×2], C4⋊D4, C2×C20 [×2], C2×C20 [×4], C5×D4 [×6], C22×C10, C22×C10 [×2], C4⋊Dic3, D6⋊C4 [×2], C2×D12, C2×C3⋊D4 [×2], C22×C12, C5×Dic3 [×2], C60 [×2], C60, S3×C10 [×6], C2×C30, C2×C30 [×2], C2×C30 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20, D4×C10 [×3], C127D4, C5×D12 [×2], C10×Dic3 [×2], C5×C3⋊D4 [×4], C2×C60 [×2], C2×C60 [×2], S3×C2×C10 [×2], C22×C30, C5×C4⋊D4, C5×C4⋊Dic3, C5×D6⋊C4 [×2], C10×D12, C10×C3⋊D4 [×2], C22×C60, C5×C127D4
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×4], C23, C10 [×7], D6 [×3], C2×D4 [×2], C4○D4, C2×C10 [×7], D12 [×2], C3⋊D4 [×2], C22×S3, C5×S3, C4⋊D4, C5×D4 [×4], C22×C10, C2×D12, C4○D12, C2×C3⋊D4, S3×C10 [×3], D4×C10 [×2], C5×C4○D4, C127D4, C5×D12 [×2], C5×C3⋊D4 [×2], S3×C2×C10, C5×C4⋊D4, C10×D12, C5×C4○D12, C10×C3⋊D4, C5×C127D4

Smallest permutation representation of C5×C127D4
On 240 points
Generators in S240
(1 20 231 63 26)(2 21 232 64 27)(3 22 233 65 28)(4 23 234 66 29)(5 24 235 67 30)(6 13 236 68 31)(7 14 237 69 32)(8 15 238 70 33)(9 16 239 71 34)(10 17 240 72 35)(11 18 229 61 36)(12 19 230 62 25)(37 206 132 169 142)(38 207 121 170 143)(39 208 122 171 144)(40 209 123 172 133)(41 210 124 173 134)(42 211 125 174 135)(43 212 126 175 136)(44 213 127 176 137)(45 214 128 177 138)(46 215 129 178 139)(47 216 130 179 140)(48 205 131 180 141)(49 156 83 224 85)(50 145 84 225 86)(51 146 73 226 87)(52 147 74 227 88)(53 148 75 228 89)(54 149 76 217 90)(55 150 77 218 91)(56 151 78 219 92)(57 152 79 220 93)(58 153 80 221 94)(59 154 81 222 95)(60 155 82 223 96)(97 193 186 160 111)(98 194 187 161 112)(99 195 188 162 113)(100 196 189 163 114)(101 197 190 164 115)(102 198 191 165 116)(103 199 192 166 117)(104 200 181 167 118)(105 201 182 168 119)(106 202 183 157 120)(107 203 184 158 109)(108 204 185 159 110)
(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 99 153 180)(2 98 154 179)(3 97 155 178)(4 108 156 177)(5 107 145 176)(6 106 146 175)(7 105 147 174)(8 104 148 173)(9 103 149 172)(10 102 150 171)(11 101 151 170)(12 100 152 169)(13 202 73 136)(14 201 74 135)(15 200 75 134)(16 199 76 133)(17 198 77 144)(18 197 78 143)(19 196 79 142)(20 195 80 141)(21 194 81 140)(22 193 82 139)(23 204 83 138)(24 203 84 137)(25 114 57 132)(26 113 58 131)(27 112 59 130)(28 111 60 129)(29 110 49 128)(30 109 50 127)(31 120 51 126)(32 119 52 125)(33 118 53 124)(34 117 54 123)(35 116 55 122)(36 115 56 121)(37 230 189 220)(38 229 190 219)(39 240 191 218)(40 239 192 217)(41 238 181 228)(42 237 182 227)(43 236 183 226)(44 235 184 225)(45 234 185 224)(46 233 186 223)(47 232 187 222)(48 231 188 221)(61 164 92 207)(62 163 93 206)(63 162 94 205)(64 161 95 216)(65 160 96 215)(66 159 85 214)(67 158 86 213)(68 157 87 212)(69 168 88 211)(70 167 89 210)(71 166 90 209)(72 165 91 208)
(1 180)(2 179)(3 178)(4 177)(5 176)(6 175)(7 174)(8 173)(9 172)(10 171)(11 170)(12 169)(13 136)(14 135)(15 134)(16 133)(17 144)(18 143)(19 142)(20 141)(21 140)(22 139)(23 138)(24 137)(25 132)(26 131)(27 130)(28 129)(29 128)(30 127)(31 126)(32 125)(33 124)(34 123)(35 122)(36 121)(37 230)(38 229)(39 240)(40 239)(41 238)(42 237)(43 236)(44 235)(45 234)(46 233)(47 232)(48 231)(49 110)(50 109)(51 120)(52 119)(53 118)(54 117)(55 116)(56 115)(57 114)(58 113)(59 112)(60 111)(61 207)(62 206)(63 205)(64 216)(65 215)(66 214)(67 213)(68 212)(69 211)(70 210)(71 209)(72 208)(73 202)(74 201)(75 200)(76 199)(77 198)(78 197)(79 196)(80 195)(81 194)(82 193)(83 204)(84 203)(85 159)(86 158)(87 157)(88 168)(89 167)(90 166)(91 165)(92 164)(93 163)(94 162)(95 161)(96 160)(97 155)(98 154)(99 153)(100 152)(101 151)(102 150)(103 149)(104 148)(105 147)(106 146)(107 145)(108 156)(181 228)(182 227)(183 226)(184 225)(185 224)(186 223)(187 222)(188 221)(189 220)(190 219)(191 218)(192 217)

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

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

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

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×C127D4 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×C127D4 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

׿
×
𝔽