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## G = C4.18D60order 480 = 25·3·5

### 3rd central extension by C4 of D60

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C4.18D60
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C60.7C4 — C4.18D60
 Lower central C15 — C30 — C60 — C4.18D60
 Upper central C1 — C4 — C2×C4 — C2×C8

Generators and relations for C4.18D60
G = < a,b,c | a4=1, b60=a2, c2=a-1, ab=ba, ac=ca, cbc-1=b59 >

Subgroups: 212 in 60 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C8, C2×C4, C10, C10, C12, C2×C6, C15, C2×C8, M4(2), C20, C2×C10, C3⋊C8, C24, C2×C12, C30, C30, C8.C4, C52C8, C40, C2×C20, C4.Dic3, C2×C24, C60, C2×C30, C4.Dic5, C2×C40, C24.C4, C153C8, C120, C2×C60, C40.6C4, C60.7C4, C2×C120, C4.18D60
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, Dic3, D6, C4⋊C4, Dic5, D10, Dic6, D12, C2×Dic3, D15, C8.C4, Dic10, D20, C2×Dic5, C4⋊Dic3, Dic15, D30, C4⋊Dic5, C24.C4, Dic30, D60, C2×Dic15, C40.6C4, C605C4, C4.18D60

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

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

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

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

126 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 5A 5B 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 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 3 4 4 4 5 5 6 6 6 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 2 1 1 2 2 2 2 2 2 2 2 2 2 60 60 60 60 2 ··· 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 2 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 C4 S3 D4 Q8 D5 Dic3 D6 Dic5 D10 D12 Dic6 D15 C8.C4 D20 Dic10 Dic15 D30 C24.C4 D60 Dic30 C40.6C4 C4.18D60 kernel C4.18D60 C60.7C4 C2×C120 C120 C2×C40 C60 C2×C30 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 1 4 1 1 1 2 2 1 4 2 2 2 4 4 4 4 8 4 8 8 8 16 32

Matrix representation of C4.18D60 in GL2(𝔽241) generated by

 64 0 0 64
,
 72 0 29 164
,
 188 23 158 53
`G:=sub<GL(2,GF(241))| [64,0,0,64],[72,29,0,164],[188,158,23,53] >;`

C4.18D60 in GAP, Magma, Sage, TeX

`C_4._{18}D_{60}`
`% in TeX`

`G:=Group("C4.18D60");`
`// GroupNames label`

`G:=SmallGroup(480,179);`
`// by ID`

`G=gap.SmallGroup(480,179);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,100,675,80,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=1,b^60=a^2,c^2=a^-1,a*b=b*a,a*c=c*a,c*b*c^-1=b^59>;`
`// generators/relations`

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