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G = C60.94D4order 480 = 25·3·5

94th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.94D4, C20.54D12, C30.15M4(2), C55(D6⋊C8), D6⋊(C52C8), (S3×C10)⋊3C8, C10.23(S3×C8), C156(C22⋊C8), C30.29(C2×C8), (C2×C20).322D6, C10.38(D6⋊C4), (C2×C12).326D10, C12.62(C5⋊D4), C4.26(C5⋊D12), C31(C20.55D4), C4.27(C15⋊D4), C20.85(C3⋊D4), C2.1(D6⋊Dic5), C10.13(C8⋊S3), C6.2(C4.Dic5), C6.1(C23.D5), C30.39(C22⋊C4), (C2×C60).224C22, (C10×Dic3).14C4, (C2×Dic3).3Dic5, C2.2(D6.Dic5), (C22×S3).2Dic5, C22.10(S3×Dic5), (S3×C2×C4).7D5, (C6×C52C8)⋊9C2, (C2×C52C8)⋊9S3, C6.4(C2×C52C8), C2.4(S3×C52C8), (S3×C2×C10).8C4, (S3×C2×C20).7C2, (C2×C153C8)⋊22C2, (C2×C10).69(C4×S3), (C2×C30).78(C2×C4), (C2×C4).227(S3×D5), (C2×C6).11(C2×Dic5), SmallGroup(480,32)

Series: Derived Chief Lower central Upper central

C1C30 — C60.94D4
C1C5C15C30C60C2×C60C6×C52C8 — C60.94D4
C15C30 — C60.94D4
C1C2×C4

Generators and relations for C60.94D4
 G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a49, cac-1=a29, cbc-1=a15b3 >

Subgroups: 316 in 100 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, S3 [×2], C6 [×3], C8 [×2], C2×C4, C2×C4 [×3], C23, C10 [×3], C10 [×2], Dic3, C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C2×C8 [×2], C22×C4, C20 [×2], C20, C2×C10, C2×C10 [×4], C3⋊C8, C24, C4×S3 [×2], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C30 [×3], C22⋊C8, C52C8 [×2], C2×C20, C2×C20 [×3], C22×C10, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C52C8, C2×C52C8, C22×C20, D6⋊C8, C3×C52C8, C153C8, S3×C20 [×2], C10×Dic3, C2×C60, S3×C2×C10, C20.55D4, C6×C52C8, C2×C153C8, S3×C2×C20, C60.94D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4 [×2], D5, D6, C22⋊C4, C2×C8, M4(2), Dic5 [×2], D10, C4×S3, D12, C3⋊D4, C22⋊C8, C52C8 [×2], C2×Dic5, C5⋊D4 [×2], S3×C8, C8⋊S3, D6⋊C4, S3×D5, C2×C52C8, C4.Dic5, C23.D5, D6⋊C8, S3×Dic5, C15⋊D4, C5⋊D12, C20.55D4, S3×C52C8, D6.Dic5, D6⋊Dic5, C60.94D4

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

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D8E8F8G8H10A···10F10G···10N12A12B12C12D15A15B20A···20H20I···20P24A···24H30A···30F60A···60H
order1222223444444556668888888810···1010···1012121212151520···2020···2024···2430···3060···60
size11116621111662222210101010303030302···26···62222442···26···610···104···44···4

84 irreducible representations

dim11111112222222222222222444444
type++++++++-+-++-+-
imageC1C2C2C2C4C4C8S3D4D5D6M4(2)Dic5D10Dic5D12C3⋊D4C4×S3C5⋊D4C52C8S3×C8C8⋊S3C4.Dic5S3×D5C15⋊D4C5⋊D12S3×Dic5S3×C52C8D6.Dic5
kernelC60.94D4C6×C52C8C2×C153C8S3×C2×C20C10×Dic3S3×C2×C10S3×C10C2×C52C8C60S3×C2×C4C2×C20C30C2×Dic3C2×C12C22×S3C20C20C2×C10C12D6C10C10C6C2×C4C4C4C22C2C2
# reps11112281221222222288448222244

Matrix representation of C60.94D4 in GL5(𝔽241)

640000
0017700
0646400
0002160
000152135
,
2330000
01638500
01567800
00012328
000228118
,
80000
01638500
01637800
00012328
000189118

G:=sub<GL(5,GF(241))| [64,0,0,0,0,0,0,64,0,0,0,177,64,0,0,0,0,0,216,152,0,0,0,0,135],[233,0,0,0,0,0,163,156,0,0,0,85,78,0,0,0,0,0,123,228,0,0,0,28,118],[8,0,0,0,0,0,163,163,0,0,0,85,78,0,0,0,0,0,123,189,0,0,0,28,118] >;

C60.94D4 in GAP, Magma, Sage, TeX

C_{60}._{94}D_4
% in TeX

G:=Group("C60.94D4");
// GroupNames label

G:=SmallGroup(480,32);
// by ID

G=gap.SmallGroup(480,32);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,1356,18822]);
// 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^15*b^3>;
// generators/relations

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