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G = C4.18D60order 480 = 25·3·5

3rd central extension by C4 of D60

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4.18D60, C120.11C4, C60.167D4, C12.36D20, C20.36D12, C8.1Dic15, C40.6Dic3, C24.1Dic5, C22.2Dic30, (C2×C40).7S3, (C2×C24).7D5, (C2×C8).5D15, (C2×C4).73D30, C30.41(C4⋊C4), (C2×C30).12Q8, C54(C24.C4), C32(C40.6C4), C60.232(C2×C4), (C2×C120).11C2, (C2×C20).398D6, C6.9(C4⋊Dic5), C4.8(C2×Dic15), C2.5(C605C4), C1510(C8.C4), C60.7C4.1C2, (C2×C12).403D10, C12.37(C2×Dic5), C20.58(C2×Dic3), (C2×C10).11Dic6, (C2×C6).11Dic10, (C2×C60).484C22, C10.16(C4⋊Dic3), SmallGroup(480,179)

Series: Derived Chief Lower central Upper central

C1C60 — C4.18D60
C1C5C15C30C60C2×C60C60.7C4 — C4.18D60
C15C30C60 — C4.18D60
C1C4C2×C4C2×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 [×2], C22, C5, C6, C6, C8 [×2], C8 [×2], C2×C4, C10, C10, C12 [×2], C2×C6, C15, C2×C8, M4(2) [×2], C20 [×2], C2×C10, C3⋊C8 [×2], C24 [×2], C2×C12, C30, C30, C8.C4, C52C8 [×2], C40 [×2], C2×C20, C4.Dic3 [×2], C2×C24, C60 [×2], C2×C30, C4.Dic5 [×2], C2×C40, C24.C4, C153C8 [×2], C120 [×2], C2×C60, C40.6C4, C60.7C4 [×2], C2×C120, C4.18D60
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, Dic3 [×2], D6, C4⋊C4, Dic5 [×2], D10, Dic6, D12, C2×Dic3, D15, C8.C4, Dic10, D20, C2×Dic5, C4⋊Dic3, Dic15 [×2], 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)
(1 146 91 176 61 206 31 236)(2 205 92 235 62 145 32 175)(3 144 93 174 63 204 33 234)(4 203 94 233 64 143 34 173)(5 142 95 172 65 202 35 232)(6 201 96 231 66 141 36 171)(7 140 97 170 67 200 37 230)(8 199 98 229 68 139 38 169)(9 138 99 168 69 198 39 228)(10 197 100 227 70 137 40 167)(11 136 101 166 71 196 41 226)(12 195 102 225 72 135 42 165)(13 134 103 164 73 194 43 224)(14 193 104 223 74 133 44 163)(15 132 105 162 75 192 45 222)(16 191 106 221 76 131 46 161)(17 130 107 160 77 190 47 220)(18 189 108 219 78 129 48 159)(19 128 109 158 79 188 49 218)(20 187 110 217 80 127 50 157)(21 126 111 156 81 186 51 216)(22 185 112 215 82 125 52 155)(23 124 113 154 83 184 53 214)(24 183 114 213 84 123 54 153)(25 122 115 152 85 182 55 212)(26 181 116 211 86 121 56 151)(27 240 117 150 87 180 57 210)(28 179 118 209 88 239 58 149)(29 238 119 148 89 178 59 208)(30 177 120 207 90 237 60 147)

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

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

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

126 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B6A6B6C8A8B8C8D8E8F8G8H10A···10F12A12B12C12D15A15B15C15D20A···20H24A···24H30A···30L40A···40P60A···60P120A···120AF
order1223444556668888888810···10121212121515151520···2024···2430···3040···4060···60120···120
size1122112222222222606060602···2222222222···22···22···22···22···22···2

126 irreducible representations

dim1111222222222222222222222
type+++++-+-+-++-++--++-
imageC1C2C2C4S3D4Q8D5Dic3D6Dic5D10D12Dic6D15C8.C4D20Dic10Dic15D30C24.C4D60Dic30C40.6C4C4.18D60
kernelC4.18D60C60.7C4C2×C120C120C2×C40C60C2×C30C2×C24C40C2×C20C24C2×C12C20C2×C10C2×C8C15C12C2×C6C8C2×C4C5C4C22C3C1
# reps121411122142224444848881632

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

640
064
,
720
29164
,
18823
15853
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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