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## G = C4⋊D60order 480 = 25·3·5

### The semidirect product of C4 and D60 acting via D60/D30=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C4⋊D60
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — C2×C4×D15 — C4⋊D60
 Lower central C15 — C2×C30 — C4⋊D60
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4⋊D60
G = < a,b,c | a4=b60=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 1476 in 188 conjugacy classes, 57 normal (33 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], C23 [×3], D5 [×4], C10 [×3], Dic3, C12 [×2], C12 [×2], D6 [×10], C2×C6, C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5, C20 [×2], C20 [×2], D10 [×10], C2×C10, C4×S3 [×2], D12 [×6], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3 [×3], D15 [×4], C30 [×3], C4⋊D4, C4×D5 [×2], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5 [×3], D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12 [×3], Dic15, C60 [×2], C60 [×2], D30 [×2], D30 [×8], C2×C30, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20 [×3], C12⋊D4, C4×D15 [×2], D60 [×6], C2×Dic15, C2×C60, C2×C60 [×2], C22×D15, C22×D15 [×2], C4⋊D20, D303C4 [×2], C15×C4⋊C4, C2×C4×D15, C2×D60, C2×D60 [×2], C4⋊D60
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], D12 [×2], C22×S3, D15, C4⋊D4, D20 [×2], C22×D5, C2×D12, S3×D4, Q83S3, D30 [×3], C2×D20, D4×D5, Q82D5, C12⋊D4, D60 [×2], C22×D15, C4⋊D20, C2×D60, D4×D15, Q83D15, C4⋊D60

Smallest permutation representation of C4⋊D60
On 240 points
Generators in S240
```(1 148 109 225)(2 226 110 149)(3 150 111 227)(4 228 112 151)(5 152 113 229)(6 230 114 153)(7 154 115 231)(8 232 116 155)(9 156 117 233)(10 234 118 157)(11 158 119 235)(12 236 120 159)(13 160 61 237)(14 238 62 161)(15 162 63 239)(16 240 64 163)(17 164 65 181)(18 182 66 165)(19 166 67 183)(20 184 68 167)(21 168 69 185)(22 186 70 169)(23 170 71 187)(24 188 72 171)(25 172 73 189)(26 190 74 173)(27 174 75 191)(28 192 76 175)(29 176 77 193)(30 194 78 177)(31 178 79 195)(32 196 80 179)(33 180 81 197)(34 198 82 121)(35 122 83 199)(36 200 84 123)(37 124 85 201)(38 202 86 125)(39 126 87 203)(40 204 88 127)(41 128 89 205)(42 206 90 129)(43 130 91 207)(44 208 92 131)(45 132 93 209)(46 210 94 133)(47 134 95 211)(48 212 96 135)(49 136 97 213)(50 214 98 137)(51 138 99 215)(52 216 100 139)(53 140 101 217)(54 218 102 141)(55 142 103 219)(56 220 104 143)(57 144 105 221)(58 222 106 145)(59 146 107 223)(60 224 108 147)
(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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(121 174)(122 173)(123 172)(124 171)(125 170)(126 169)(127 168)(128 167)(129 166)(130 165)(131 164)(132 163)(133 162)(134 161)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(175 180)(176 179)(177 178)(181 208)(182 207)(183 206)(184 205)(185 204)(186 203)(187 202)(188 201)(189 200)(190 199)(191 198)(192 197)(193 196)(194 195)(209 240)(210 239)(211 238)(212 237)(213 236)(214 235)(215 234)(216 233)(217 232)(218 231)(219 230)(220 229)(221 228)(222 227)(223 226)(224 225)```

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 C4○D4 D10 D12 D15 D20 D30 D60 S3×D4 Q8⋊3S3 D4×D5 Q8⋊2D5 D4×D15 Q8⋊3D15 kernel C4⋊D60 D30⋊3C4 C15×C4⋊C4 C2×C4×D15 C2×D60 C5×C4⋊C4 C60 D30 C3×C4⋊C4 C2×C20 C30 C2×C12 C20 C4⋊C4 C12 C2×C4 C4 C10 C10 C6 C6 C2 C2 # reps 1 2 1 1 3 1 2 2 2 3 2 6 4 4 8 12 16 1 1 2 2 4 4

Matrix representation of C4⋊D60 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 22 34 0 0 0 0 27 39
,
 27 2 0 0 0 0 23 4 0 0 0 0 0 0 53 23 0 0 0 0 38 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 36 32 0 0 0 0 11 25 0 0 0 0 0 0 53 23 0 0 0 0 45 8 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,22,27,0,0,0,0,34,39],[27,23,0,0,0,0,2,4,0,0,0,0,0,0,53,38,0,0,0,0,23,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[36,11,0,0,0,0,32,25,0,0,0,0,0,0,53,45,0,0,0,0,23,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C4⋊D60 in GAP, Magma, Sage, TeX

`C_4\rtimes D_{60}`
`% in TeX`

`G:=Group("C4:D60");`
`// GroupNames label`

`G:=SmallGroup(480,860);`
`// by ID`

`G=gap.SmallGroup(480,860);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,58,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^60=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

׿
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