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## G = C6×C5⋊2C8order 240 = 24·3·5

### Direct product of C6 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C6×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C60 — C3×C5⋊2C8 — C6×C5⋊2C8
 Lower central C5 — C6×C5⋊2C8
 Upper central C1 — C2×C12

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

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

96 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5A 5B 6A ··· 6F 8A ··· 8H 10A ··· 10F 12A ··· 12H 15A 15B 15C 15D 20A ··· 20H 24A ··· 24P 30A ··· 30L 60A ··· 60P order 1 2 2 2 3 3 4 4 4 4 5 5 6 ··· 6 8 ··· 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 1 1 1 1 1 1 2 2 1 ··· 1 5 ··· 5 2 ··· 2 1 ··· 1 2 2 2 2 2 ··· 2 5 ··· 5 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 D5 Dic5 D10 Dic5 C3×D5 C5⋊2C8 C3×Dic5 C6×D5 C3×Dic5 C3×C5⋊2C8 kernel C6×C5⋊2C8 C3×C5⋊2C8 C2×C60 C2×C5⋊2C8 C60 C2×C30 C5⋊2C8 C2×C20 C30 C20 C2×C10 C10 C2×C12 C12 C12 C2×C6 C2×C4 C6 C4 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 2 2 2 2 4 8 4 4 4 16

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

 240 0 0 0 16 0 0 0 16
,
 1 0 0 0 51 240 0 1 0
,
 1 0 0 0 34 12 0 59 207
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

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