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G = C60.D4order 480 = 25·3·5

106th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.106D4, (C2×C30).3Q8, C153C8.7C4, C20.44(C4×S3), C12.12(C4×D5), (C2×C20).63D6, C30.32(C4⋊C4), C158(C8.C4), C60.128(C2×C4), (C2×C12).63D10, (C2×C6).1Dic10, (C2×C10).2Dic6, C4.Dic3.4D5, C4.Dic5.4S3, C22.3(C15⋊Q8), C20.89(C3⋊D4), C12.89(C5⋊D4), C32(C20.53D4), C54(C12.53D4), C4.31(C15⋊D4), (C2×C60).212C22, C4.13(D30.C2), C6.8(C10.D4), C2.5(Dic155C4), C10.14(Dic3⋊C4), (C2×C4).192(S3×D5), (C2×C153C8).16C2, (C5×C4.Dic3).4C2, (C3×C4.Dic5).4C2, SmallGroup(480,68)

Series: Derived Chief Lower central Upper central

C1C60 — C60.D4
C1C5C15C30C60C2×C60C3×C4.Dic5 — C60.D4
C15C30C60 — C60.D4
C1C4C2×C4

Generators and relations for C60.D4
 G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a29, cac-1=a49, cbc-1=b3 >

Subgroups: 188 in 60 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4 [×2], C22, C5, C6, C6, C8 [×4], C2×C4, C10, C10, C12 [×2], C2×C6, C15, C2×C8, M4(2) [×2], C20 [×2], C2×C10, C3⋊C8 [×3], C24, C2×C12, C30, C30, C8.C4, C52C8 [×3], C40, C2×C20, C2×C3⋊C8, C4.Dic3, C3×M4(2), C60 [×2], C2×C30, C2×C52C8, C4.Dic5, C5×M4(2), C12.53D4, C5×C3⋊C8, C3×C52C8, C153C8 [×2], C2×C60, C20.53D4, C3×C4.Dic5, C5×C4.Dic3, C2×C153C8, C60.D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, D10, Dic6, C4×S3, C3⋊D4, C8.C4, Dic10, C4×D5, C5⋊D4, Dic3⋊C4, S3×D5, C10.D4, C12.53D4, D30.C2, C15⋊D4, C15⋊Q8, C20.53D4, Dic155C4, C60.D4

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

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

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

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

60 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B6A6B8A8B8C8D8E8F8G8H10A10B10C10D12A12B12C15A15B20A20B20C20D20E20F24A24B24C24D30A···30F40A···40H60A···60H
order12234445566888888881010101012121215152020202020202424242430···3040···4060···60
size112211222241212202030303030224422444222244202020204···412···124···4

60 irreducible representations

dim1111122222222222224444444
type++++++-+++--++--
imageC1C2C2C2C4S3D4Q8D5D6D10C4×S3C3⋊D4Dic6C8.C4C4×D5C5⋊D4Dic10S3×D5C12.53D4D30.C2C15⋊D4C15⋊Q8C20.53D4C60.D4
kernelC60.D4C3×C4.Dic5C5×C4.Dic3C2×C153C8C153C8C4.Dic5C60C2×C30C4.Dic3C2×C20C2×C12C20C20C2×C10C15C12C12C2×C6C2×C4C5C4C4C22C3C1
# reps1111411121222244442222248

Matrix representation of C60.D4 in GL6(𝔽241)

6400000
0640000
0005100
001895200
00001240
000010
,
3000000
080000
001025600
0018013900
00009216
0000225232
,
02110000
3000000
001436200
00209800
00009943
0000198142

G:=sub<GL(6,GF(241))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,0,189,0,0,0,0,51,52,0,0,0,0,0,0,1,1,0,0,0,0,240,0],[30,0,0,0,0,0,0,8,0,0,0,0,0,0,102,180,0,0,0,0,56,139,0,0,0,0,0,0,9,225,0,0,0,0,216,232],[0,30,0,0,0,0,211,0,0,0,0,0,0,0,143,20,0,0,0,0,62,98,0,0,0,0,0,0,99,198,0,0,0,0,43,142] >;

C60.D4 in GAP, Magma, Sage, TeX

C_{60}.D_4
% in TeX

G:=Group("C60.D4");
// GroupNames label

G:=SmallGroup(480,68);
// by ID

G=gap.SmallGroup(480,68);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,36,100,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45,b*a*b^-1=a^29,c*a*c^-1=a^49,c*b*c^-1=b^3>;
// generators/relations

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