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G = C8.6D30order 480 = 25·3·5

3rd non-split extension by C8 of D30 acting via D30/D15=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.5D4, C8.6D30, Q161D15, C30.41D8, C40.14D6, C1513SD32, D120.3C2, C24.14D10, C120.11C22, (C5×Q16)⋊1S3, (C3×Q16)⋊1D5, C153C163C2, (C15×Q16)⋊1C2, C53(C8.6D6), C33(C5⋊SD32), C2.6(D4⋊D15), C6.19(D4⋊D5), C4.3(C157D4), C10.19(D4⋊S3), C12.19(C5⋊D4), C20.17(C3⋊D4), SmallGroup(480,188)

Series: Derived Chief Lower central Upper central

C1C120 — C8.6D30
C1C5C15C30C60C120D120 — C8.6D30
C15C30C60C120 — C8.6D30
C1C2C4C8Q16

Generators and relations for C8.6D30
 G = < a,b,c | a8=1, b30=a4, c2=a3, bab-1=a-1, ac=ca, cbc-1=a-1b29 >

120C2
4C4
60C22
40S3
24D5
2Q8
30D4
4C12
20D6
4C20
12D10
8D15
15D8
15C16
2C3×Q8
10D12
2C5×Q8
6D20
4D30
4C60
15SD32
5C3⋊C16
5D24
3D40
3C52C16
2Q8×C15
2D60
5C8.6D6
3C5⋊SD32

Smallest permutation representation of C8.6D30
On 240 points
Generators in S240
(1 220 117 134 31 190 87 164)(2 165 88 191 32 135 118 221)(3 222 119 136 33 192 89 166)(4 167 90 193 34 137 120 223)(5 224 61 138 35 194 91 168)(6 169 92 195 36 139 62 225)(7 226 63 140 37 196 93 170)(8 171 94 197 38 141 64 227)(9 228 65 142 39 198 95 172)(10 173 96 199 40 143 66 229)(11 230 67 144 41 200 97 174)(12 175 98 201 42 145 68 231)(13 232 69 146 43 202 99 176)(14 177 100 203 44 147 70 233)(15 234 71 148 45 204 101 178)(16 179 102 205 46 149 72 235)(17 236 73 150 47 206 103 180)(18 121 104 207 48 151 74 237)(19 238 75 152 49 208 105 122)(20 123 106 209 50 153 76 239)(21 240 77 154 51 210 107 124)(22 125 108 211 52 155 78 181)(23 182 79 156 53 212 109 126)(24 127 110 213 54 157 80 183)(25 184 81 158 55 214 111 128)(26 129 112 215 56 159 82 185)(27 186 83 160 57 216 113 130)(28 131 114 217 58 161 84 187)(29 188 85 162 59 218 115 132)(30 133 116 219 60 163 86 189)
(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 30 134 219 87 86 220 133 31 60 164 189 117 116 190 163)(2 162 191 115 118 188 165 59 32 132 221 85 88 218 135 29)(3 28 136 217 89 84 222 131 33 58 166 187 119 114 192 161)(4 160 193 113 120 186 167 57 34 130 223 83 90 216 137 27)(5 26 138 215 91 82 224 129 35 56 168 185 61 112 194 159)(6 158 195 111 62 184 169 55 36 128 225 81 92 214 139 25)(7 24 140 213 93 80 226 127 37 54 170 183 63 110 196 157)(8 156 197 109 64 182 171 53 38 126 227 79 94 212 141 23)(9 22 142 211 95 78 228 125 39 52 172 181 65 108 198 155)(10 154 199 107 66 240 173 51 40 124 229 77 96 210 143 21)(11 20 144 209 97 76 230 123 41 50 174 239 67 106 200 153)(12 152 201 105 68 238 175 49 42 122 231 75 98 208 145 19)(13 18 146 207 99 74 232 121 43 48 176 237 69 104 202 151)(14 150 203 103 70 236 177 47 44 180 233 73 100 206 147 17)(15 16 148 205 101 72 234 179 45 46 178 235 71 102 204 149)

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

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

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

60 conjugacy classes

class 1 2A2B 3 4A4B5A5B 6 8A8B10A10B12A12B12C15A15B15C15D16A16B16C16D20A20B20C20D20E20F24A24B30A30B30C30D40A40B40C40D60A60B60C60D60E···60L120A···120H
order1223445568810101212121515151516161616202020202020242430303030404040406060606060···60120···120
size111202282222222488222230303030448888442222444444448···84···4

60 irreducible representations

dim1111222222222222444444
type++++++++++++++++++
imageC1C2C2C2S3D4D5D6D8D10C3⋊D4D15SD32C5⋊D4D30C157D4D4⋊S3D4⋊D5C8.6D6C5⋊SD32D4⋊D15C8.6D30
kernelC8.6D30C153C16D120C15×Q16C5×Q16C60C3×Q16C40C30C24C20Q16C15C12C8C4C10C6C5C3C2C1
# reps1111112122244448122448

Matrix representation of C8.6D30 in GL4(𝔽241) generated by

240000
024000
0023011
00230230
,
1109800
108200
00103200
00200138
,
693300
10417200
0010341
00200103
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,230,230,0,0,11,230],[110,108,0,0,98,2,0,0,0,0,103,200,0,0,200,138],[69,104,0,0,33,172,0,0,0,0,103,200,0,0,41,103] >;

C8.6D30 in GAP, Magma, Sage, TeX

C_8._6D_{30}
% in TeX

G:=Group("C8.6D30");
// GroupNames label

G:=SmallGroup(480,188);
// by ID

G=gap.SmallGroup(480,188);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,120,254,135,142,675,346,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^30=a^4,c^2=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^29>;
// generators/relations

Export

Subgroup lattice of C8.6D30 in TeX

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