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G = C60.54D4order 480 = 25·3·5

54th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.54D4, C20.6D12, (C2×C20).44D6, (C2×C12).45D10, C12.9(C5⋊D4), C20.7(C3⋊D4), (C2×Dic3).Dic5, (C2×Dic6).5D5, C4.Dic5.2S3, C10.40(D6⋊C4), C60.7C4.2C2, C4.20(C5⋊D12), C4.13(C15⋊D4), C155(C4.10D4), C54(C12.47D4), C31(C20.10D4), C2.4(D6⋊Dic5), (C2×C60).23C22, (C10×Dic3).1C4, (C10×Dic6).1C2, C6.3(C23.D5), C22.4(S3×Dic5), C30.45(C22⋊C4), (C2×C4).5(S3×D5), (C2×C30).84(C2×C4), (C2×C10).71(C4×S3), (C2×C6).2(C2×Dic5), (C3×C4.Dic5).1C2, SmallGroup(480,38)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.54D4
C1C5C15C30C60C2×C60C3×C4.Dic5 — C60.54D4
C15C30C2×C30 — C60.54D4
C1C2C2×C4

Generators and relations for C60.54D4
 G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a-1, cac-1=a49, cbc-1=a45b3 >

Subgroups: 252 in 76 conjugacy classes, 34 normal (30 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) [×2], C2×Q8, C20 [×2], C20 [×2], C2×C10, C3⋊C8, C24, Dic6 [×2], C2×Dic3 [×2], C2×C12, C30, C30, C4.10D4, C52C8 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C4.Dic3, C3×M4(2), C2×Dic6, C5×Dic3 [×2], C60 [×2], C2×C30, C4.Dic5, C4.Dic5, Q8×C10, C12.47D4, C3×C52C8, C153C8, C5×Dic6 [×2], C10×Dic3 [×2], C2×C60, C20.10D4, C3×C4.Dic5, C60.7C4, C10×Dic6, C60.54D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, C4.10D4, C2×Dic5, C5⋊D4 [×2], D6⋊C4, S3×D5, C23.D5, C12.47D4, S3×Dic5, C15⋊D4, C5⋊D12, C20.10D4, D6⋊Dic5, C60.54D4

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

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

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

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

57 conjugacy classes

class 1 2A2B 3 4A4B4C4D5A5B6A6B8A8B8C8D10A···10F12A12B12C15A15B20A20B20C20D20E···20L24A24B24C24D30A···30F60A···60H
order122344445566888810···1012121215152020202020···202424242430···3060···60
size11222212122224202060602···222444444412···12202020204···44···4

57 irreducible representations

dim11111222222222244444444
type++++++++-++-+--+-
imageC1C2C2C2C4S3D4D5D6Dic5D10D12C3⋊D4C4×S3C5⋊D4C4.10D4S3×D5C12.47D4C15⋊D4C5⋊D12S3×Dic5C20.10D4C60.54D4
kernelC60.54D4C3×C4.Dic5C60.7C4C10×Dic6C10×Dic3C4.Dic5C60C2×Dic6C2×C20C2×Dic3C2×C12C20C20C2×C10C12C15C2×C4C5C4C4C22C3C1
# reps11114122142222812222248

Matrix representation of C60.54D4 in GL6(𝔽241)

22600000
210160000
00872600
001115400
0095240205
001024360
,
168730000
135730000
00330179207
0017601210
00921554154
001621654154
,
17700000
113640000
001410280
00002401
001311000
009901000

G:=sub<GL(6,GF(241))| [226,210,0,0,0,0,0,16,0,0,0,0,0,0,87,11,95,10,0,0,26,154,24,24,0,0,0,0,0,36,0,0,0,0,205,0],[168,135,0,0,0,0,73,73,0,0,0,0,0,0,33,176,92,162,0,0,0,0,15,16,0,0,179,1,54,54,0,0,207,210,154,154],[177,113,0,0,0,0,0,64,0,0,0,0,0,0,141,0,13,99,0,0,0,0,1,0,0,0,28,240,100,100,0,0,0,1,0,0] >;

C60.54D4 in GAP, Magma, Sage, TeX

C_{60}._{54}D_4
% in TeX

G:=Group("C60.54D4");
// GroupNames label

G:=SmallGroup(480,38);
// by ID

G=gap.SmallGroup(480,38);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,100,675,1356,18822]);
// Polycyclic

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

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