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G = C5×C12.C8order 480 = 25·3·5

Direct product of C5 and C12.C8

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C12.C8, C60.11C8, C12.1C40, C24.5C20, C40.83D6, C120.22C4, C1515M5(2), C40.11Dic3, C120.110C22, C3⋊C165C10, C20.7(C3⋊C8), (C2×C6).3C40, C6.9(C2×C40), C32(C5×M5(2)), C8.21(S3×C10), (C2×C40).17S3, C30.71(C2×C8), (C2×C30).11C8, (C2×C60).45C4, C8.2(C5×Dic3), (C2×C24).13C10, (C2×C120).33C2, C60.254(C2×C4), C12.39(C2×C20), C24.26(C2×C10), (C2×C12).10C20, C20.72(C2×Dic3), (C2×C20).25Dic3, C4.11(C10×Dic3), C4.(C5×C3⋊C8), C22.(C5×C3⋊C8), C2.4(C10×C3⋊C8), (C5×C3⋊C16)⋊12C2, C10.23(C2×C3⋊C8), (C2×C8).7(C5×S3), (C2×C10).3(C3⋊C8), (C2×C4).5(C5×Dic3), SmallGroup(480,131)

Series: Derived Chief Lower central Upper central

C1C6 — C5×C12.C8
C1C3C6C12C24C120C5×C3⋊C16 — C5×C12.C8
C3C6 — C5×C12.C8
C1C40C2×C40

Generators and relations for C5×C12.C8
 G = < a,b,c | a5=b24=1, c4=b18, ab=ba, ac=ca, cbc-1=b5 >

2C2
2C6
2C10
2C30
3C16
3C16
3M5(2)
3C80
3C80
3C5×M5(2)

Smallest permutation representation of C5×C12.C8
On 240 points
Generators in S240
(1 104 95 61 33)(2 105 96 62 34)(3 106 73 63 35)(4 107 74 64 36)(5 108 75 65 37)(6 109 76 66 38)(7 110 77 67 39)(8 111 78 68 40)(9 112 79 69 41)(10 113 80 70 42)(11 114 81 71 43)(12 115 82 72 44)(13 116 83 49 45)(14 117 84 50 46)(15 118 85 51 47)(16 119 86 52 48)(17 120 87 53 25)(18 97 88 54 26)(19 98 89 55 27)(20 99 90 56 28)(21 100 91 57 29)(22 101 92 58 30)(23 102 93 59 31)(24 103 94 60 32)(121 235 211 178 148)(122 236 212 179 149)(123 237 213 180 150)(124 238 214 181 151)(125 239 215 182 152)(126 240 216 183 153)(127 217 193 184 154)(128 218 194 185 155)(129 219 195 186 156)(130 220 196 187 157)(131 221 197 188 158)(132 222 198 189 159)(133 223 199 190 160)(134 224 200 191 161)(135 225 201 192 162)(136 226 202 169 163)(137 227 203 170 164)(138 228 204 171 165)(139 229 205 172 166)(140 230 206 173 167)(141 231 207 174 168)(142 232 208 175 145)(143 233 209 176 146)(144 234 210 177 147)
(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)
(1 133 10 130 19 127 4 124 13 121 22 142 7 139 16 136)(2 138 11 135 20 132 5 129 14 126 23 123 8 144 17 141)(3 143 12 140 21 137 6 134 15 131 24 128 9 125 18 122)(25 168 34 165 43 162 28 159 37 156 46 153 31 150 40 147)(26 149 35 146 44 167 29 164 38 161 47 158 32 155 41 152)(27 154 36 151 45 148 30 145 39 166 48 163 33 160 42 157)(49 178 58 175 67 172 52 169 61 190 70 187 55 184 64 181)(50 183 59 180 68 177 53 174 62 171 71 192 56 189 65 186)(51 188 60 185 69 182 54 179 63 176 72 173 57 170 66 191)(73 209 82 206 91 203 76 200 85 197 94 194 79 215 88 212)(74 214 83 211 92 208 77 205 86 202 95 199 80 196 89 193)(75 195 84 216 93 213 78 210 87 207 96 204 81 201 90 198)(97 236 106 233 115 230 100 227 109 224 118 221 103 218 112 239)(98 217 107 238 116 235 101 232 110 229 119 226 104 223 113 220)(99 222 108 219 117 240 102 237 111 234 120 231 105 228 114 225)

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

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

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

180 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B5C5D6A6B6C8A8B8C8D8E8F10A10B10C10D10E10F10G10H12A12B12C12D15A15B15C15D16A···16H20A···20H20I20J20K20L24A···24H30A···30L40A···40P40Q···40X60A···60P80A···80AF120A···120AF
order122344455556668888881010101010101010121212121515151516···1620···202020202024···2430···3040···4040···4060···6080···80120···120
size1122112111122211112211112222222222226···61···122222···22···21···12···22···26···62···2

180 irreducible representations

dim111111111111112222222222222222
type++++-+-
imageC1C2C2C4C4C5C8C8C10C10C20C20C40C40S3Dic3D6Dic3C3⋊C8C3⋊C8C5×S3M5(2)C5×Dic3S3×C10C5×Dic3C12.C8C5×C3⋊C8C5×C3⋊C8C5×M5(2)C5×C12.C8
kernelC5×C12.C8C5×C3⋊C16C2×C120C120C2×C60C12.C8C60C2×C30C3⋊C16C2×C24C24C2×C12C12C2×C6C2×C40C40C40C2×C20C20C2×C10C2×C8C15C8C8C2×C4C5C4C22C3C1
# reps1212244484881616111122444448881632

Matrix representation of C5×C12.C8 in GL2(𝔽241) generated by

980
098
,
2390
0209
,
01
2110
G:=sub<GL(2,GF(241))| [98,0,0,98],[239,0,0,209],[0,211,1,0] >;

C5×C12.C8 in GAP, Magma, Sage, TeX

C_5\times C_{12}.C_8
% in TeX

G:=Group("C5xC12.C8");
// GroupNames label

G:=SmallGroup(480,131);
// by ID

G=gap.SmallGroup(480,131);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,1149,80,102,15686]);
// Polycyclic

G:=Group<a,b,c|a^5=b^24=1,c^4=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C5×C12.C8 in TeX

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