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## G = C4×C15⋊7D4order 480 = 25·3·5

### Direct product of C4 and C15⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C4×C15⋊7D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — C2×C15⋊7D4 — C4×C15⋊7D4
 Lower central C15 — C30 — C4×C15⋊7D4
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C4×C157D4
G = < a,b,c,d | a4=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 948 in 188 conjugacy classes, 71 normal (47 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×6], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], Dic3 [×4], C12 [×2], C12, D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, Dic5 [×4], C20 [×2], C20, D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C22×S3, C22×C6, D15 [×2], C30 [×3], C30 [×2], C4×D4, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, Dic15 [×2], Dic15 [×2], C60 [×2], C60, D30 [×2], D30 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, C4×C3⋊D4, C4×D15 [×2], C2×Dic15 [×3], C157D4 [×4], C2×C60 [×2], C2×C60 [×2], C22×D15, C22×C30, C4×C5⋊D4, C4×Dic15, C30.4Q8, D303C4, C30.38D4, C2×C4×D15, C2×C157D4, C22×C60, C4×C157D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], C3⋊D4 [×2], C22×S3, D15, C4×D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, S3×C2×C4, C4○D12, C2×C3⋊D4, D30 [×3], C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C3⋊D4, C4×D15 [×2], C157D4 [×2], C22×D15, C4×C5⋊D4, C2×C4×D15, D6011C2, C2×C157D4, C4×C157D4

Smallest permutation representation of C4×C157D4
On 240 points
Generators in S240
(1 99 41 66)(2 100 42 67)(3 101 43 68)(4 102 44 69)(5 103 45 70)(6 104 31 71)(7 105 32 72)(8 91 33 73)(9 92 34 74)(10 93 35 75)(11 94 36 61)(12 95 37 62)(13 96 38 63)(14 97 39 64)(15 98 40 65)(16 117 47 83)(17 118 48 84)(18 119 49 85)(19 120 50 86)(20 106 51 87)(21 107 52 88)(22 108 53 89)(23 109 54 90)(24 110 55 76)(25 111 56 77)(26 112 57 78)(27 113 58 79)(28 114 59 80)(29 115 60 81)(30 116 46 82)(121 225 162 190)(122 211 163 191)(123 212 164 192)(124 213 165 193)(125 214 151 194)(126 215 152 195)(127 216 153 181)(128 217 154 182)(129 218 155 183)(130 219 156 184)(131 220 157 185)(132 221 158 186)(133 222 159 187)(134 223 160 188)(135 224 161 189)(136 231 171 197)(137 232 172 198)(138 233 173 199)(139 234 174 200)(140 235 175 201)(141 236 176 202)(142 237 177 203)(143 238 178 204)(144 239 179 205)(145 240 180 206)(146 226 166 207)(147 227 167 208)(148 228 168 209)(149 229 169 210)(150 230 170 196)
(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)
(1 156 16 180)(2 155 17 179)(3 154 18 178)(4 153 19 177)(5 152 20 176)(6 151 21 175)(7 165 22 174)(8 164 23 173)(9 163 24 172)(10 162 25 171)(11 161 26 170)(12 160 27 169)(13 159 28 168)(14 158 29 167)(15 157 30 166)(31 125 52 140)(32 124 53 139)(33 123 54 138)(34 122 55 137)(35 121 56 136)(36 135 57 150)(37 134 58 149)(38 133 59 148)(39 132 60 147)(40 131 46 146)(41 130 47 145)(42 129 48 144)(43 128 49 143)(44 127 50 142)(45 126 51 141)(61 224 78 230)(62 223 79 229)(63 222 80 228)(64 221 81 227)(65 220 82 226)(66 219 83 240)(67 218 84 239)(68 217 85 238)(69 216 86 237)(70 215 87 236)(71 214 88 235)(72 213 89 234)(73 212 90 233)(74 211 76 232)(75 225 77 231)(91 192 109 199)(92 191 110 198)(93 190 111 197)(94 189 112 196)(95 188 113 210)(96 187 114 209)(97 186 115 208)(98 185 116 207)(99 184 117 206)(100 183 118 205)(101 182 119 204)(102 181 120 203)(103 195 106 202)(104 194 107 201)(105 193 108 200)
(1 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 45)(13 44)(14 43)(15 42)(16 47)(17 46)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 105)(76 109)(77 108)(78 107)(79 106)(80 120)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 110)(121 174)(122 173)(123 172)(124 171)(125 170)(126 169)(127 168)(128 167)(129 166)(130 180)(131 179)(132 178)(133 177)(134 176)(135 175)(136 165)(137 164)(138 163)(139 162)(140 161)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)(181 228)(182 227)(183 226)(184 240)(185 239)(186 238)(187 237)(188 236)(189 235)(190 234)(191 233)(192 232)(193 231)(194 230)(195 229)(196 214)(197 213)(198 212)(199 211)(200 225)(201 224)(202 223)(203 222)(204 221)(205 220)(206 219)(207 218)(208 217)(209 216)(210 215)

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

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

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

132 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G ··· 4L 5A 5B 6A ··· 6G 10A ··· 10N 12A ··· 12H 15A 15B 15C 15D 20A ··· 20P 30A ··· 30AB 60A ··· 60AF order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 30 30 2 1 1 1 1 2 2 30 ··· 30 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

132 irreducible representations

 dim 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 2 2 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D5 D6 D6 C4○D4 D10 D10 C3⋊D4 C4×S3 D15 C5⋊D4 C4×D5 C4○D12 D30 D30 C4○D20 C15⋊7D4 C4×D15 D60⋊11C2 kernel C4×C15⋊7D4 C4×Dic15 C30.4Q8 D30⋊3C4 C30.38D4 C2×C4×D15 C2×C15⋊7D4 C22×C60 C15⋊7D4 C22×C20 C60 C22×C12 C2×C20 C22×C10 C30 C2×C12 C22×C6 C20 C2×C10 C22×C4 C12 C2×C6 C10 C2×C4 C23 C6 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 2 1 2 4 2 4 4 4 8 8 4 8 4 8 16 16 16

Matrix representation of C4×C157D4 in GL4(𝔽61) generated by

 50 0 0 0 0 50 0 0 0 0 60 0 0 0 0 60
,
 23 39 0 0 14 45 0 0 0 0 0 44 0 0 18 43
,
 43 43 0 0 1 18 0 0 0 0 8 14 0 0 52 53
,
 18 18 0 0 60 43 0 0 0 0 43 1 0 0 43 18
G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,60,0,0,0,0,60],[23,14,0,0,39,45,0,0,0,0,0,18,0,0,44,43],[43,1,0,0,43,18,0,0,0,0,8,52,0,0,14,53],[18,60,0,0,18,43,0,0,0,0,43,43,0,0,1,18] >;

C4×C157D4 in GAP, Magma, Sage, TeX

C_4\times C_{15}\rtimes_7D_4
% in TeX

G:=Group("C4xC15:7D4");
// GroupNames label

G:=SmallGroup(480,893);
// by ID

G=gap.SmallGroup(480,893);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,58,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^15=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