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G = C60.105D4order 480 = 25·3·5

105th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.105D4, C20.59D12, (C2×C30).2Q8, C12.70(C4×D5), (C2×C20).62D6, C30.31(C4⋊C4), C53(C24.C4), C157(C8.C4), C60.113(C2×C4), C4.13(D5×Dic3), (C2×C6).5Dic10, (C2×C10).1Dic6, C52C8.1Dic3, C4.Dic3.1D5, C22.2(C15⋊Q8), C60.7C4.5C2, (C2×C12).314D10, C4.31(C5⋊D12), C31(C20.53D4), C12.67(C5⋊D4), (C2×C60).45C22, C20.27(C2×Dic3), C10.11(C4⋊Dic3), C6.7(C10.D4), C2.5(C30.Q8), (C3×C52C8).1C4, (C6×C52C8).1C2, (C2×C52C8).5S3, (C2×C4).141(S3×D5), (C5×C4.Dic3).2C2, SmallGroup(480,67)

Series: Derived Chief Lower central Upper central

C1C60 — C60.105D4
C1C5C15C30C60C2×C60C6×C52C8 — C60.105D4
C15C30C60 — C60.105D4
C1C4C2×C4

Generators and relations for C60.105D4
 G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a49, cac-1=a29, cbc-1=a30b3 >

Subgroups: 188 in 60 conjugacy classes, 34 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 [×2], C24 [×2], C2×C12, C30, C30, C8.C4, C52C8 [×2], C52C8, C40, C2×C20, C4.Dic3, C4.Dic3, C2×C24, C60 [×2], C2×C30, C2×C52C8, C4.Dic5, C5×M4(2), C24.C4, C5×C3⋊C8, C3×C52C8 [×2], C153C8, C2×C60, C20.53D4, C6×C52C8, C5×C4.Dic3, C60.7C4, C60.105D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, Dic3 [×2], D6, C4⋊C4, D10, Dic6, D12, C2×Dic3, C8.C4, Dic10, C4×D5, C5⋊D4, C4⋊Dic3, S3×D5, C10.D4, C24.C4, D5×Dic3, C5⋊D12, C15⋊Q8, C20.53D4, C30.Q8, C60.105D4

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

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

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

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

66 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B6A6B6C8A8B8C8D8E8F8G8H10A10B10C10D12A12B12C12D15A15B20A20B20C20D20E20F24A···24H30A···30F40A···40H60A···60H
order122344455666888888881010101012121212151520202020202024···2430···3040···4060···60
size1122112222221010101012126060224422224422224410···104···412···124···4

66 irreducible representations

dim1111122222222222222444444
type++++++-+-+++--+-+-
imageC1C2C2C2C4S3D4Q8D5Dic3D6D10D12Dic6C8.C4C4×D5C5⋊D4Dic10C24.C4S3×D5D5×Dic3C5⋊D12C15⋊Q8C20.53D4C60.105D4
kernelC60.105D4C6×C52C8C5×C4.Dic3C60.7C4C3×C52C8C2×C52C8C60C2×C30C4.Dic3C52C8C2×C20C2×C12C20C2×C10C15C12C12C2×C6C5C2×C4C4C4C22C3C1
# reps1111411122122244448222248

Matrix representation of C60.105D4 in GL4(𝔽241) generated by

181000
023700
005151
001901
,
211000
0800
0063154
00234178
,
0800
8000
0010080
00119141
G:=sub<GL(4,GF(241))| [181,0,0,0,0,237,0,0,0,0,51,190,0,0,51,1],[211,0,0,0,0,8,0,0,0,0,63,234,0,0,154,178],[0,8,0,0,8,0,0,0,0,0,100,119,0,0,80,141] >;

C60.105D4 in GAP, Magma, Sage, TeX

C_{60}._{105}D_4
% in TeX

G:=Group("C60.105D4");
// GroupNames label

G:=SmallGroup(480,67);
// by ID

G=gap.SmallGroup(480,67);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,36,100,346,80,1356,18822]);
// Polycyclic

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

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