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## G = C6×C5⋊C8order 240 = 24·3·5

### Direct product of C6 and C5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C6×C5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C3×Dic5 — C3×C5⋊C8 — C6×C5⋊C8
 Lower central C5 — C6×C5⋊C8
 Upper central C1 — C2×C6

Generators and relations for C6×C5⋊C8
G = < a,b,c | a6=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C6×C5⋊C8
Regular action on 240 points
Generators in S240
(1 235 101 55 145 189)(2 236 102 56 146 190)(3 237 103 49 147 191)(4 238 104 50 148 192)(5 239 97 51 149 185)(6 240 98 52 150 186)(7 233 99 53 151 187)(8 234 100 54 152 188)(9 21 57 159 111 203)(10 22 58 160 112 204)(11 23 59 153 105 205)(12 24 60 154 106 206)(13 17 61 155 107 207)(14 18 62 156 108 208)(15 19 63 157 109 201)(16 20 64 158 110 202)(25 69 79 115 215 165)(26 70 80 116 216 166)(27 71 73 117 209 167)(28 72 74 118 210 168)(29 65 75 119 211 161)(30 66 76 120 212 162)(31 67 77 113 213 163)(32 68 78 114 214 164)(33 197 169 217 85 121)(34 198 170 218 86 122)(35 199 171 219 87 123)(36 200 172 220 88 124)(37 193 173 221 81 125)(38 194 174 222 82 126)(39 195 175 223 83 127)(40 196 176 224 84 128)(41 225 95 131 137 181)(42 226 96 132 138 182)(43 227 89 133 139 183)(44 228 90 134 140 184)(45 229 91 135 141 177)(46 230 92 136 142 178)(47 231 93 129 143 179)(48 232 94 130 144 180)
(1 19 77 35 137)(2 36 20 138 78)(3 139 37 79 21)(4 80 140 22 38)(5 23 73 39 141)(6 40 24 142 74)(7 143 33 75 17)(8 76 144 18 34)(9 191 133 125 69)(10 126 192 70 134)(11 71 127 135 185)(12 136 72 186 128)(13 187 129 121 65)(14 122 188 66 130)(15 67 123 131 189)(16 132 68 190 124)(25 203 147 89 81)(26 90 204 82 148)(27 83 91 149 205)(28 150 84 206 92)(29 207 151 93 85)(30 94 208 86 152)(31 87 95 145 201)(32 146 88 202 96)(41 101 157 213 171)(42 214 102 172 158)(43 173 215 159 103)(44 160 174 104 216)(45 97 153 209 175)(46 210 98 176 154)(47 169 211 155 99)(48 156 170 100 212)(49 227 221 165 111)(50 166 228 112 222)(51 105 167 223 229)(52 224 106 230 168)(53 231 217 161 107)(54 162 232 108 218)(55 109 163 219 225)(56 220 110 226 164)(57 237 183 193 115)(58 194 238 116 184)(59 117 195 177 239)(60 178 118 240 196)(61 233 179 197 119)(62 198 234 120 180)(63 113 199 181 235)(64 182 114 236 200)
(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)| (1,235,101,55,145,189)(2,236,102,56,146,190)(3,237,103,49,147,191)(4,238,104,50,148,192)(5,239,97,51,149,185)(6,240,98,52,150,186)(7,233,99,53,151,187)(8,234,100,54,152,188)(9,21,57,159,111,203)(10,22,58,160,112,204)(11,23,59,153,105,205)(12,24,60,154,106,206)(13,17,61,155,107,207)(14,18,62,156,108,208)(15,19,63,157,109,201)(16,20,64,158,110,202)(25,69,79,115,215,165)(26,70,80,116,216,166)(27,71,73,117,209,167)(28,72,74,118,210,168)(29,65,75,119,211,161)(30,66,76,120,212,162)(31,67,77,113,213,163)(32,68,78,114,214,164)(33,197,169,217,85,121)(34,198,170,218,86,122)(35,199,171,219,87,123)(36,200,172,220,88,124)(37,193,173,221,81,125)(38,194,174,222,82,126)(39,195,175,223,83,127)(40,196,176,224,84,128)(41,225,95,131,137,181)(42,226,96,132,138,182)(43,227,89,133,139,183)(44,228,90,134,140,184)(45,229,91,135,141,177)(46,230,92,136,142,178)(47,231,93,129,143,179)(48,232,94,130,144,180), (1,19,77,35,137)(2,36,20,138,78)(3,139,37,79,21)(4,80,140,22,38)(5,23,73,39,141)(6,40,24,142,74)(7,143,33,75,17)(8,76,144,18,34)(9,191,133,125,69)(10,126,192,70,134)(11,71,127,135,185)(12,136,72,186,128)(13,187,129,121,65)(14,122,188,66,130)(15,67,123,131,189)(16,132,68,190,124)(25,203,147,89,81)(26,90,204,82,148)(27,83,91,149,205)(28,150,84,206,92)(29,207,151,93,85)(30,94,208,86,152)(31,87,95,145,201)(32,146,88,202,96)(41,101,157,213,171)(42,214,102,172,158)(43,173,215,159,103)(44,160,174,104,216)(45,97,153,209,175)(46,210,98,176,154)(47,169,211,155,99)(48,156,170,100,212)(49,227,221,165,111)(50,166,228,112,222)(51,105,167,223,229)(52,224,106,230,168)(53,231,217,161,107)(54,162,232,108,218)(55,109,163,219,225)(56,220,110,226,164)(57,237,183,193,115)(58,194,238,116,184)(59,117,195,177,239)(60,178,118,240,196)(61,233,179,197,119)(62,198,234,120,180)(63,113,199,181,235)(64,182,114,236,200), (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( (1,235,101,55,145,189)(2,236,102,56,146,190)(3,237,103,49,147,191)(4,238,104,50,148,192)(5,239,97,51,149,185)(6,240,98,52,150,186)(7,233,99,53,151,187)(8,234,100,54,152,188)(9,21,57,159,111,203)(10,22,58,160,112,204)(11,23,59,153,105,205)(12,24,60,154,106,206)(13,17,61,155,107,207)(14,18,62,156,108,208)(15,19,63,157,109,201)(16,20,64,158,110,202)(25,69,79,115,215,165)(26,70,80,116,216,166)(27,71,73,117,209,167)(28,72,74,118,210,168)(29,65,75,119,211,161)(30,66,76,120,212,162)(31,67,77,113,213,163)(32,68,78,114,214,164)(33,197,169,217,85,121)(34,198,170,218,86,122)(35,199,171,219,87,123)(36,200,172,220,88,124)(37,193,173,221,81,125)(38,194,174,222,82,126)(39,195,175,223,83,127)(40,196,176,224,84,128)(41,225,95,131,137,181)(42,226,96,132,138,182)(43,227,89,133,139,183)(44,228,90,134,140,184)(45,229,91,135,141,177)(46,230,92,136,142,178)(47,231,93,129,143,179)(48,232,94,130,144,180), (1,19,77,35,137)(2,36,20,138,78)(3,139,37,79,21)(4,80,140,22,38)(5,23,73,39,141)(6,40,24,142,74)(7,143,33,75,17)(8,76,144,18,34)(9,191,133,125,69)(10,126,192,70,134)(11,71,127,135,185)(12,136,72,186,128)(13,187,129,121,65)(14,122,188,66,130)(15,67,123,131,189)(16,132,68,190,124)(25,203,147,89,81)(26,90,204,82,148)(27,83,91,149,205)(28,150,84,206,92)(29,207,151,93,85)(30,94,208,86,152)(31,87,95,145,201)(32,146,88,202,96)(41,101,157,213,171)(42,214,102,172,158)(43,173,215,159,103)(44,160,174,104,216)(45,97,153,209,175)(46,210,98,176,154)(47,169,211,155,99)(48,156,170,100,212)(49,227,221,165,111)(50,166,228,112,222)(51,105,167,223,229)(52,224,106,230,168)(53,231,217,161,107)(54,162,232,108,218)(55,109,163,219,225)(56,220,110,226,164)(57,237,183,193,115)(58,194,238,116,184)(59,117,195,177,239)(60,178,118,240,196)(61,233,179,197,119)(62,198,234,120,180)(63,113,199,181,235)(64,182,114,236,200), (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([[(1,235,101,55,145,189),(2,236,102,56,146,190),(3,237,103,49,147,191),(4,238,104,50,148,192),(5,239,97,51,149,185),(6,240,98,52,150,186),(7,233,99,53,151,187),(8,234,100,54,152,188),(9,21,57,159,111,203),(10,22,58,160,112,204),(11,23,59,153,105,205),(12,24,60,154,106,206),(13,17,61,155,107,207),(14,18,62,156,108,208),(15,19,63,157,109,201),(16,20,64,158,110,202),(25,69,79,115,215,165),(26,70,80,116,216,166),(27,71,73,117,209,167),(28,72,74,118,210,168),(29,65,75,119,211,161),(30,66,76,120,212,162),(31,67,77,113,213,163),(32,68,78,114,214,164),(33,197,169,217,85,121),(34,198,170,218,86,122),(35,199,171,219,87,123),(36,200,172,220,88,124),(37,193,173,221,81,125),(38,194,174,222,82,126),(39,195,175,223,83,127),(40,196,176,224,84,128),(41,225,95,131,137,181),(42,226,96,132,138,182),(43,227,89,133,139,183),(44,228,90,134,140,184),(45,229,91,135,141,177),(46,230,92,136,142,178),(47,231,93,129,143,179),(48,232,94,130,144,180)], [(1,19,77,35,137),(2,36,20,138,78),(3,139,37,79,21),(4,80,140,22,38),(5,23,73,39,141),(6,40,24,142,74),(7,143,33,75,17),(8,76,144,18,34),(9,191,133,125,69),(10,126,192,70,134),(11,71,127,135,185),(12,136,72,186,128),(13,187,129,121,65),(14,122,188,66,130),(15,67,123,131,189),(16,132,68,190,124),(25,203,147,89,81),(26,90,204,82,148),(27,83,91,149,205),(28,150,84,206,92),(29,207,151,93,85),(30,94,208,86,152),(31,87,95,145,201),(32,146,88,202,96),(41,101,157,213,171),(42,214,102,172,158),(43,173,215,159,103),(44,160,174,104,216),(45,97,153,209,175),(46,210,98,176,154),(47,169,211,155,99),(48,156,170,100,212),(49,227,221,165,111),(50,166,228,112,222),(51,105,167,223,229),(52,224,106,230,168),(53,231,217,161,107),(54,162,232,108,218),(55,109,163,219,225),(56,220,110,226,164),(57,237,183,193,115),(58,194,238,116,184),(59,117,195,177,239),(60,178,118,240,196),(61,233,179,197,119),(62,198,234,120,180),(63,113,199,181,235),(64,182,114,236,200)], [(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×C5⋊C8 is a maximal subgroup of   C30.M4(2)  Dic5.22D12  D30⋊C8  C30.4M4(2)  Dic15⋊C8  C5⋊C8.D6

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 F5 C5⋊C8 C2×F5 C3×F5 C3×C5⋊C8 C6×F5 kernel C6×C5⋊C8 C3×C5⋊C8 C6×Dic5 C2×C5⋊C8 C3×Dic5 C2×C30 C5⋊C8 C2×Dic5 C30 Dic5 C2×C10 C10 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 1 2 1 2 4 2

Matrix representation of C6×C5⋊C8 in GL5(𝔽241)

 240 0 0 0 0 0 225 0 0 0 0 0 225 0 0 0 0 0 225 0 0 0 0 0 225
,
 1 0 0 0 0 0 0 0 0 240 0 1 0 0 240 0 0 1 0 240 0 0 0 1 240
,
 1 0 0 0 0 0 58 214 213 6 0 30 220 204 64 0 21 37 177 36 0 235 9 183 27

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,225,0,0,0,0,0,225,0,0,0,0,0,225,0,0,0,0,0,225],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,240,240,240,240],[1,0,0,0,0,0,58,30,21,235,0,214,220,37,9,0,213,204,177,183,0,6,64,36,27] >;

C6×C5⋊C8 in GAP, Magma, Sage, TeX

C_6\times C_5\rtimes C_8
% in TeX

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

G:=SmallGroup(240,115);
// by ID

G=gap.SmallGroup(240,115);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,69,3461,599]);
// 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^3>;
// generators/relations

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