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G = C60.210D4order 480 = 25·3·5

10th non-split extension by C60 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.210D4, M4(2).1D15, C22.1Dic30, (C2×C30).1Q8, C153C8.5C4, C20.49(C4×S3), C60.79(C2×C4), (C2×C20).68D6, C4.13(C4×D15), (C2×C4).36D30, C12.17(C4×D5), C30.42(C4⋊C4), (C2×C12).69D10, (C2×C10).3Dic6, (C2×C6).3Dic10, C1513(C8.C4), C60.7C4.6C2, C33(C20.53D4), C55(C12.53D4), C4.28(C157D4), (C2×C60).54C22, (C5×M4(2)).3S3, (C3×M4(2)).3D5, C20.107(C3⋊D4), C12.107(C5⋊D4), (C15×M4(2)).5C2, C2.5(C30.4Q8), C10.20(Dic3⋊C4), C6.13(C10.D4), (C2×C153C8).4C2, SmallGroup(480,182)

Series: Derived Chief Lower central Upper central

C1C60 — C60.210D4
C1C5C15C30C60C2×C60C2×C153C8 — C60.210D4
C15C30C60 — C60.210D4
C1C4C2×C4M4(2)

Generators and relations for C60.210D4
 G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=cac-1=a29, cbc-1=a30b3 >

Subgroups: 212 in 60 conjugacy classes, 33 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), M4(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, C153C8 [×2], C153C8, C120, C2×C60, C20.53D4, C2×C153C8, C60.7C4, C15×M4(2), C60.210D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, D10, Dic6, C4×S3, C3⋊D4, D15, C8.C4, Dic10, C4×D5, C5⋊D4, Dic3⋊C4, D30, C10.D4, C12.53D4, Dic30, C4×D15, C157D4, C20.53D4, C30.4Q8, C60.210D4

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

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

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

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

84 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B6A6B8A8B8C8D8E8F8G8H10A10B10C10D12A12B12C15A15B15C15D20A20B20C20D20E20F24A24B24C24D30A30B30C30D30E30F30G30H40A···40H60A···60H60I60J60K60L120A···120P
order1223444556688888888101010101212121515151520202020202024242424303030303030303040···4060···6060606060120···120
size1122112222444303030306060224422422222222444444222244444···42···244444···4

84 irreducible representations

dim11111222222222222222222444
type++++++-+++-+-+-
imageC1C2C2C2C4S3D4Q8D5D6D10C4×S3C3⋊D4Dic6D15C8.C4C4×D5C5⋊D4Dic10D30C4×D15C157D4Dic30C12.53D4C20.53D4C60.210D4
kernelC60.210D4C2×C153C8C60.7C4C15×M4(2)C153C8C5×M4(2)C60C2×C30C3×M4(2)C2×C20C2×C12C20C20C2×C10M4(2)C15C12C12C2×C6C2×C4C4C4C22C5C3C1
# reps11114111212222444444888248

Matrix representation of C60.210D4 in GL4(𝔽241) generated by

801600
7715700
00640
00064
,
14313800
99800
00300
00189233
,
14313800
99800
00550
00102236
G:=sub<GL(4,GF(241))| [80,77,0,0,16,157,0,0,0,0,64,0,0,0,0,64],[143,9,0,0,138,98,0,0,0,0,30,189,0,0,0,233],[143,9,0,0,138,98,0,0,0,0,5,102,0,0,50,236] >;

C60.210D4 in GAP, Magma, Sage, TeX

C_{60}._{210}D_4
% in TeX

G:=Group("C60.210D4");
// GroupNames label

G:=SmallGroup(480,182);
// by ID

G=gap.SmallGroup(480,182);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,36,100,346,80,2693,18822]);
// Polycyclic

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

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