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G = C4.D60order 480 = 25·3·5

4th non-split extension by C4 of D60 acting via D60/D30=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4.12D60, C20.12D12, C60.204D4, C12.12D20, M4(2).2D15, (C2×C4).2D30, (C2×C20).70D6, (C2×C12).71D10, C22.5(C4×D15), C10.35(D6⋊C4), C60.7C4.7C2, C157(C4.10D4), C53(C12.47D4), C4.22(C157D4), C32(C4.12D20), (C2×C60).56C22, (C2×Dic15).1C4, (C2×Dic30).7C2, (C5×M4(2)).4S3, (C3×M4(2)).4D5, C12.101(C5⋊D4), C20.101(C3⋊D4), C30.77(C22⋊C4), (C15×M4(2)).6C2, C6.20(D10⋊C4), C2.10(D303C4), (C2×C6).6(C4×D5), (C2×C30).66(C2×C4), (C2×C10).29(C4×S3), SmallGroup(480,184)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C4.D60
C1C5C15C30C60C2×C60C2×Dic30 — C4.D60
C15C30C2×C30 — C4.D60
C1C2C2×C4M4(2)

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

Subgroups: 404 in 76 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C3, C4 [×2], C4 [×2], C22, C5, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C10, C10, Dic3 [×2], C12 [×2], C2×C6, C15, M4(2), M4(2), C2×Q8, Dic5 [×2], C20 [×2], C2×C10, C3⋊C8, C24, Dic6 [×2], C2×Dic3 [×2], C2×C12, C30, C30, C4.10D4, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×C20, C4.Dic3, C3×M4(2), C2×Dic6, Dic15 [×2], C60 [×2], C2×C30, C4.Dic5, C5×M4(2), C2×Dic10, C12.47D4, C153C8, C120, Dic30 [×2], C2×Dic15 [×2], C2×C60, C4.12D20, C60.7C4, C15×M4(2), C2×Dic30, C4.D60
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C4.10D4, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, C12.47D4, C4×D15, D60, C157D4, C4.12D20, D303C4, C4.D60

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

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

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

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

81 conjugacy classes

class 1 2A2B 3 4A4B4C4D5A5B6A6B8A8B8C8D10A10B10C10D12A12B12C15A15B15C15D20A20B20C20D20E20F24A24B24C24D30A30B30C30D30E30F30G30H40A···40H60A···60H60I60J60K60L120A···120P
order1223444455668888101010101212121515151520202020202024242424303030303030303040···4060···6060606060120···120
size11222260602224446060224422422222222444444222244444···42···244444···4

81 irreducible representations

dim1111122222222222222224444
type++++++++++++++----
imageC1C2C2C2C4S3D4D5D6D10D12C3⋊D4C4×S3D15D20C5⋊D4C4×D5D30D60C157D4C4×D15C4.10D4C12.47D4C4.12D20C4.D60
kernelC4.D60C60.7C4C15×M4(2)C2×Dic30C2×Dic15C5×M4(2)C60C3×M4(2)C2×C20C2×C12C20C20C2×C10M4(2)C12C12C2×C6C2×C4C4C4C22C15C5C3C1
# reps1111412212222444448881248

Matrix representation of C4.D60 in GL6(𝔽241)

100000
010000
0011515300
0010112600
00181699198
0017218743142
,
107190000
2221070000
008981236236
0082722170
00237129101102
00151205102220
,
2071100000
110340000
00196191207115
0097382833
00441822309
002418418

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,115,101,18,172,0,0,153,126,16,187,0,0,0,0,99,43,0,0,0,0,198,142],[107,222,0,0,0,0,19,107,0,0,0,0,0,0,89,82,237,151,0,0,81,72,129,205,0,0,236,21,101,102,0,0,236,70,102,220],[207,110,0,0,0,0,110,34,0,0,0,0,0,0,196,97,44,24,0,0,191,38,182,18,0,0,207,28,230,4,0,0,115,33,9,18] >;

C4.D60 in GAP, Magma, Sage, TeX

C_4.D_{60}
% in TeX

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

G:=SmallGroup(480,184);
// by ID

G=gap.SmallGroup(480,184);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,36,422,100,346,2693,18822]);
// Polycyclic

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

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×
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