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## G = D60.6C4order 480 = 25·3·5

### The non-split extension by D60 of C4 acting through Inn(D60)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D60.6C4
 Chief series C1 — C5 — C15 — C30 — C60 — C4×D15 — D60⋊11C2 — D60.6C4
 Lower central C15 — C30 — D60.6C4
 Upper central C1 — C8 — C2×C8

Generators and relations for D60.6C4
G = < a,b,c | a60=b2=1, c4=a30, bab=a-1, ac=ca, bc=cb >

Subgroups: 596 in 124 conjugacy classes, 55 normal (41 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, C2×C8 [×2], M4(2) [×3], 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], C4.Dic3, C2×C24, C4○D12, Dic15 [×2], C60 [×2], D30 [×2], C2×C30, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C2×C40, C4○D20, C8○D12, C153C8 [×2], C120 [×2], Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, D20.3C4, C8×D15 [×2], C40⋊S3 [×2], C60.7C4, C2×C120, D6011C2, D60.6C4
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, C8○D12, C4×D15 [×2], C22×D15, D20.3C4, C2×C4×D15, D60.6C4

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

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

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

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

132 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A ··· 10F 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 2 3 4 4 4 4 4 5 5 6 6 6 8 8 8 8 8 8 8 8 8 8 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 2 30 30 2 1 1 2 30 30 2 2 2 2 2 1 1 1 1 2 2 30 30 30 30 2 ··· 2 2 2 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 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 C8○D12 C4×D15 C4×D15 D20.3C4 D60.6C4 kernel D60.6C4 C8×D15 C40⋊S3 C60.7C4 C2×C120 D60⋊11C2 Dic30 D60 C15⋊7D4 C2×C40 C2×C24 C40 C2×C20 C24 C2×C12 C20 C2×C10 C2×C8 C15 C12 C2×C6 C8 C2×C4 C5 C4 C22 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 8 16 32

Matrix representation of D60.6C4 in GL4(𝔽241) generated by

 131 80 0 0 161 148 0 0 0 0 185 44 0 0 115 0
,
 157 147 0 0 93 84 0 0 0 0 0 197 0 0 115 0
,
 177 0 0 0 0 177 0 0 0 0 8 0 0 0 0 8
`G:=sub<GL(4,GF(241))| [131,161,0,0,80,148,0,0,0,0,185,115,0,0,44,0],[157,93,0,0,147,84,0,0,0,0,0,115,0,0,197,0],[177,0,0,0,0,177,0,0,0,0,8,0,0,0,0,8] >;`

D60.6C4 in GAP, Magma, Sage, TeX

`D_{60}._6C_4`
`% in TeX`

`G:=Group("D60.6C4");`
`// GroupNames label`

`G:=SmallGroup(480,866);`
`// by ID`

`G=gap.SmallGroup(480,866);`
`# by ID`

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

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

׿
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