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G = C60.7C8order 480 = 25·3·5

1st non-split extension by C60 of C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C60.7C8, C40.73D6, C8.22D30, C120.14C4, C24.78D10, C1513M5(2), C40.9Dic3, C24.5Dic5, C8.2Dic15, C120.91C22, C20.4(C3⋊C8), C4.(C153C8), (C2×C30).7C8, C153C168C2, (C2×C8).7D15, C30.61(C2×C8), (C2×C60).37C4, (C2×C40).10S3, (C2×C24).13D5, C54(C12.C8), C32(C20.4C8), C12.1(C52C8), C60.235(C2×C4), (C2×C120).17C2, C22.(C153C8), (C2×C4).5Dic15, (C2×C12).8Dic5, C4.10(C2×Dic15), C20.61(C2×Dic3), C12.40(C2×Dic5), (C2×C20).19Dic3, C10.18(C2×C3⋊C8), C6.9(C2×C52C8), C2.4(C2×C153C8), (C2×C10).5(C3⋊C8), (C2×C6).3(C52C8), SmallGroup(480,172)

Series: Derived Chief Lower central Upper central

C1C30 — C60.7C8
C1C5C15C30C60C120C153C16 — C60.7C8
C15C30 — C60.7C8
C1C8C2×C8

Generators and relations for C60.7C8
 G = < a,b | a60=1, b8=a30, bab-1=a-1 >

2C2
2C6
2C10
2C30
15C16
15C16
15M5(2)
5C3⋊C16
5C3⋊C16
3C52C16
3C52C16
5C12.C8
3C20.4C8

Smallest permutation representation of C60.7C8
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)
(1 233 83 159 46 188 68 174 31 203 113 129 16 218 98 144)(2 232 84 158 47 187 69 173 32 202 114 128 17 217 99 143)(3 231 85 157 48 186 70 172 33 201 115 127 18 216 100 142)(4 230 86 156 49 185 71 171 34 200 116 126 19 215 101 141)(5 229 87 155 50 184 72 170 35 199 117 125 20 214 102 140)(6 228 88 154 51 183 73 169 36 198 118 124 21 213 103 139)(7 227 89 153 52 182 74 168 37 197 119 123 22 212 104 138)(8 226 90 152 53 181 75 167 38 196 120 122 23 211 105 137)(9 225 91 151 54 240 76 166 39 195 61 121 24 210 106 136)(10 224 92 150 55 239 77 165 40 194 62 180 25 209 107 135)(11 223 93 149 56 238 78 164 41 193 63 179 26 208 108 134)(12 222 94 148 57 237 79 163 42 192 64 178 27 207 109 133)(13 221 95 147 58 236 80 162 43 191 65 177 28 206 110 132)(14 220 96 146 59 235 81 161 44 190 66 176 29 205 111 131)(15 219 97 145 60 234 82 160 45 189 67 175 30 204 112 130)

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

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

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

132 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B6A6B6C8A8B8C8D8E8F10A···10F12A12B12C12D15A15B15C15D16A···16H20A···20H24A···24H30A···30L40A···40P60A···60P120A···120AF
order12234445566688888810···10121212121515151516···1620···2024···2430···3040···4060···60120···120
size1122112222221111222···22222222230···302···22···22···22···22···22···2

132 irreducible representations

dim11111112222222222222222222222
type+++++-+--+-+-+-
imageC1C2C2C4C4C8C8S3D5Dic3D6Dic3Dic5D10Dic5C3⋊C8C3⋊C8D15M5(2)C52C8C52C8Dic15D30Dic15C12.C8C153C8C153C8C20.4C8C60.7C8
kernelC60.7C8C153C16C2×C120C120C2×C60C60C2×C30C2×C40C2×C24C40C40C2×C20C24C24C2×C12C20C2×C10C2×C8C15C12C2×C6C8C8C2×C4C5C4C22C3C1
# reps1212244121112222244444448881632

Matrix representation of C60.7C8 in GL2(𝔽241) generated by

1580
090
,
01
2110
G:=sub<GL(2,GF(241))| [158,0,0,90],[0,211,1,0] >;

C60.7C8 in GAP, Magma, Sage, TeX

C_{60}._7C_8
% in TeX

G:=Group("C60.7C8");
// GroupNames label

G:=SmallGroup(480,172);
// by ID

G=gap.SmallGroup(480,172);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,58,80,2693,18822]);
// Polycyclic

G:=Group<a,b|a^60=1,b^8=a^30,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C60.7C8 in TeX

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