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G = C6×C52C8order 240 = 24·3·5

Direct product of C6 and C52C8

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

Aliases: C6×C52C8, C304C8, C102C24, C60.13C4, C20.6C12, C12.58D10, C12.6Dic5, C60.71C22, C54(C2×C24), C1514(C2×C8), (C2×C20).6C6, (C2×C30).8C4, C4.14(C6×D5), C20.15(C2×C6), (C2×C10).4C12, C30.54(C2×C4), (C2×C60).16C2, (C2×C12).11D5, C2.1(C6×Dic5), (C2×C6).4Dic5, C4.3(C3×Dic5), C10.12(C2×C12), C6.11(C2×Dic5), C22.2(C3×Dic5), (C2×C4).5(C3×D5), SmallGroup(240,38)

Series: Derived Chief Lower central Upper central

C1C5 — C6×C52C8
C1C5C10C20C60C3×C52C8 — C6×C52C8
C5 — C6×C52C8
C1C2×C12

Generators and relations for C6×C52C8
 G = < a,b,c | a6=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

5C8
5C8
5C2×C8
5C24
5C24
5C2×C24

Smallest permutation representation of C6×C52C8
Regular action on 240 points
Generators in S240
(1 164 83 205 180 44)(2 165 84 206 181 45)(3 166 85 207 182 46)(4 167 86 208 183 47)(5 168 87 201 184 48)(6 161 88 202 177 41)(7 162 81 203 178 42)(8 163 82 204 179 43)(9 150 71 111 231 32)(10 151 72 112 232 25)(11 152 65 105 225 26)(12 145 66 106 226 27)(13 146 67 107 227 28)(14 147 68 108 228 29)(15 148 69 109 229 30)(16 149 70 110 230 31)(17 236 218 123 104 137)(18 237 219 124 97 138)(19 238 220 125 98 139)(20 239 221 126 99 140)(21 240 222 127 100 141)(22 233 223 128 101 142)(23 234 224 121 102 143)(24 235 217 122 103 144)(33 188 91 80 120 52)(34 189 92 73 113 53)(35 190 93 74 114 54)(36 191 94 75 115 55)(37 192 95 76 116 56)(38 185 96 77 117 49)(39 186 89 78 118 50)(40 187 90 79 119 51)(57 213 193 156 132 172)(58 214 194 157 133 173)(59 215 195 158 134 174)(60 216 196 159 135 175)(61 209 197 160 136 176)(62 210 198 153 129 169)(63 211 199 154 130 170)(64 212 200 155 131 171)
(1 50 124 231 59)(2 60 232 125 51)(3 52 126 225 61)(4 62 226 127 53)(5 54 128 227 63)(6 64 228 121 55)(7 56 122 229 57)(8 58 230 123 49)(9 195 83 186 138)(10 139 187 84 196)(11 197 85 188 140)(12 141 189 86 198)(13 199 87 190 142)(14 143 191 88 200)(15 193 81 192 144)(16 137 185 82 194)(17 96 204 157 149)(18 150 158 205 89)(19 90 206 159 151)(20 152 160 207 91)(21 92 208 153 145)(22 146 154 201 93)(23 94 202 155 147)(24 148 156 203 95)(25 98 40 165 216)(26 209 166 33 99)(27 100 34 167 210)(28 211 168 35 101)(29 102 36 161 212)(30 213 162 37 103)(31 104 38 163 214)(32 215 164 39 97)(41 171 108 224 115)(42 116 217 109 172)(43 173 110 218 117)(44 118 219 111 174)(45 175 112 220 119)(46 120 221 105 176)(47 169 106 222 113)(48 114 223 107 170)(65 136 182 80 239)(66 240 73 183 129)(67 130 184 74 233)(68 234 75 177 131)(69 132 178 76 235)(70 236 77 179 133)(71 134 180 78 237)(72 238 79 181 135)
(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)

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

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

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

C6×C52C8 is a maximal subgroup of
Dic154C8  C30.22C42  C30.23C42  C60.94D4  D304C8  C10.D24  D6015C4  C10.Dic12  Dic3015C4  C60.14Q8  C60.15Q8  C60.7Q8  C60.8Q8  C60.105D4  Dic5×C24  C60.C8  D12.2Dic5  D60.5C4  C20.60D12  D5×C2×C24

96 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D5A5B6A···6F8A···8H10A···10F12A···12H15A15B15C15D20A···20H24A···24P30A···30L60A···60P
order1222334444556···68···810···1012···121515151520···2024···2430···3060···60
size1111111111221···15···52···21···122222···25···52···22···2

96 irreducible representations

dim1111111111112222222222
type++++-+-
imageC1C2C2C3C4C4C6C6C8C12C12C24D5Dic5D10Dic5C3×D5C52C8C3×Dic5C6×D5C3×Dic5C3×C52C8
kernelC6×C52C8C3×C52C8C2×C60C2×C52C8C60C2×C30C52C8C2×C20C30C20C2×C10C10C2×C12C12C12C2×C6C2×C4C6C4C4C22C2
# reps121222428441622224844416

Matrix representation of C6×C52C8 in GL3(𝔽241) generated by

24000
0160
0016
,
100
051240
010
,
100
03412
059207
G:=sub<GL(3,GF(241))| [240,0,0,0,16,0,0,0,16],[1,0,0,0,51,1,0,240,0],[1,0,0,0,34,59,0,12,207] >;

C6×C52C8 in GAP, Magma, Sage, TeX

C_6\times C_5\rtimes_2C_8
% in TeX

G:=Group("C6xC5:2C8");
// GroupNames label

G:=SmallGroup(240,38);
// by ID

G=gap.SmallGroup(240,38);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^6=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×C52C8 in TeX

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