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## G = D60.3C4order 480 = 25·3·5

### 1st non-split extension by D60 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D60.3C4
 Chief series C1 — C5 — C15 — C30 — C60 — C4×D15 — D60⋊11C2 — D60.3C4
 Lower central C15 — C30 — D60.3C4
 Upper central C1 — C4 — M4(2)

Generators and relations for D60.3C4
G = < a,b,c | a60=b2=1, c4=a30, bab=a-1, cac-1=a31, cbc-1=a30b >

Subgroups: 596 in 124 conjugacy classes, 55 normal (35 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3 [×2], C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, D5 [×2], C10, C10, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C15, C2×C8 [×3], M4(2), M4(2) [×2], C4○D4, Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, D15 [×2], C30, C30, C8○D4, C52C8 [×2], C40 [×2], Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8, C3×M4(2), C4○D12, Dic15 [×2], C60 [×2], D30 [×2], C2×C30, C8×D5 [×2], C8⋊D5 [×2], C2×C52C8, C5×M4(2), C4○D20, D12.C4, C153C8 [×2], C120 [×2], Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, D20.2C4, C8×D15 [×2], C40⋊S3 [×2], C2×C153C8, C15×M4(2), D6011C2, D60.3C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, D15, C8○D4, C4×D5 [×2], C22×D5, S3×C2×C4, D30 [×3], C2×C4×D5, D12.C4, C4×D15 [×2], C22×D15, D20.2C4, C2×C4×D15, D60.3C4

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

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A 10B 10C 10D 12A 12B 12C 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40H 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P order 1 2 2 2 2 3 4 4 4 4 4 5 5 6 6 8 8 8 8 8 8 8 8 8 8 10 10 10 10 12 12 12 15 15 15 15 20 20 20 20 20 20 24 24 24 24 30 30 30 30 30 30 30 30 40 ··· 40 60 ··· 60 60 60 60 60 120 ··· 120 size 1 1 2 30 30 2 1 1 2 30 30 2 2 2 4 2 2 2 2 15 15 15 15 30 30 2 2 4 4 2 2 4 2 2 2 2 2 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4 2 ··· 2 4 4 4 4 4 ··· 4

90 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 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D5 D6 D6 D10 D10 C4×S3 C4×S3 D15 C8○D4 C4×D5 C4×D5 D30 D30 C4×D15 C4×D15 D12.C4 D20.2C4 D60.3C4 kernel D60.3C4 C8×D15 C40⋊S3 C2×C15⋊3C8 C15×M4(2) D60⋊11C2 Dic30 D60 C15⋊7D4 C5×M4(2) C3×M4(2) C40 C2×C20 C24 C2×C12 C20 C2×C10 M4(2) C15 C12 C2×C6 C8 C2×C4 C4 C22 C5 C3 C1 # reps 1 2 2 1 1 1 2 2 4 1 2 2 1 4 2 2 2 4 4 4 4 8 4 8 8 2 4 8

Matrix representation of D60.3C4 in GL4(𝔽241) generated by

 16 131 0 0 46 211 0 0 0 0 64 128 0 0 0 177
,
 178 147 0 0 114 63 0 0 0 0 64 128 0 0 177 177
,
 64 0 0 0 0 64 0 0 0 0 30 0 0 0 211 211
`G:=sub<GL(4,GF(241))| [16,46,0,0,131,211,0,0,0,0,64,0,0,0,128,177],[178,114,0,0,147,63,0,0,0,0,64,177,0,0,128,177],[64,0,0,0,0,64,0,0,0,0,30,211,0,0,0,211] >;`

D60.3C4 in GAP, Magma, Sage, TeX

`D_{60}._3C_4`
`% in TeX`

`G:=Group("D60.3C4");`
`// GroupNames label`

`G:=SmallGroup(480,872);`
`// by ID`

`G=gap.SmallGroup(480,872);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,219,58,80,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^2=1,c^4=a^30,b*a*b=a^-1,c*a*c^-1=a^31,c*b*c^-1=a^30*b>;`
`// generators/relations`

׿
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