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## G = C2×C15⋊3C8order 240 = 24·3·5

### Direct product of C2 and C15⋊3C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×C15⋊3C8
 Chief series C1 — C5 — C15 — C30 — C60 — C15⋊3C8 — C2×C15⋊3C8
 Lower central C15 — C2×C15⋊3C8
 Upper central C1 — C2×C4

Generators and relations for C2×C153C8
G = < a,b,c | a2=b15=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C2×C153C8
Regular action on 240 points
Generators in S240
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 61)(16 90)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 84)(26 85)(27 86)(28 87)(29 88)(30 89)(31 102)(32 103)(33 104)(34 105)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 106)(59 107)(60 108)(121 181)(122 182)(123 183)(124 184)(125 185)(126 186)(127 187)(128 188)(129 189)(130 190)(131 191)(132 192)(133 193)(134 194)(135 195)(136 210)(137 196)(138 197)(139 198)(140 199)(141 200)(142 201)(143 202)(144 203)(145 204)(146 205)(147 206)(148 207)(149 208)(150 209)(151 215)(152 216)(153 217)(154 218)(155 219)(156 220)(157 221)(158 222)(159 223)(160 224)(161 225)(162 211)(163 212)(164 213)(165 214)(166 240)(167 226)(168 227)(169 228)(170 229)(171 230)(172 231)(173 232)(174 233)(175 234)(176 235)(177 236)(178 237)(179 238)(180 239)
(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)
(1 226 54 196 18 212 42 193)(2 240 55 210 19 211 43 192)(3 239 56 209 20 225 44 191)(4 238 57 208 21 224 45 190)(5 237 58 207 22 223 31 189)(6 236 59 206 23 222 32 188)(7 235 60 205 24 221 33 187)(8 234 46 204 25 220 34 186)(9 233 47 203 26 219 35 185)(10 232 48 202 27 218 36 184)(11 231 49 201 28 217 37 183)(12 230 50 200 29 216 38 182)(13 229 51 199 30 215 39 181)(14 228 52 198 16 214 40 195)(15 227 53 197 17 213 41 194)(61 168 116 138 76 164 97 134)(62 167 117 137 77 163 98 133)(63 166 118 136 78 162 99 132)(64 180 119 150 79 161 100 131)(65 179 120 149 80 160 101 130)(66 178 106 148 81 159 102 129)(67 177 107 147 82 158 103 128)(68 176 108 146 83 157 104 127)(69 175 109 145 84 156 105 126)(70 174 110 144 85 155 91 125)(71 173 111 143 86 154 92 124)(72 172 112 142 87 153 93 123)(73 171 113 141 88 152 94 122)(74 170 114 140 89 151 95 121)(75 169 115 139 90 165 96 135)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - - + - + - + - image C1 C2 C2 C4 C4 C8 S3 D5 Dic3 D6 Dic3 Dic5 D10 Dic5 C3⋊C8 D15 C5⋊2C8 Dic15 D30 Dic15 C15⋊3C8 kernel C2×C15⋊3C8 C15⋊3C8 C2×C60 C60 C2×C30 C30 C2×C20 C2×C12 C20 C20 C2×C10 C12 C12 C2×C6 C10 C2×C4 C6 C4 C4 C22 C2 # reps 1 2 1 2 2 8 1 2 1 1 1 2 2 2 4 4 8 4 4 4 16

Matrix representation of C2×C153C8 in GL5(𝔽241)

 1 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 52 52 0 0 0 189 240 0 0 0 0 0 225 131 0 0 0 46 30
,
 8 0 0 0 0 0 35 2 0 0 0 110 206 0 0 0 0 0 184 177 0 0 0 111 57

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,52,189,0,0,0,52,240,0,0,0,0,0,225,46,0,0,0,131,30],[8,0,0,0,0,0,35,110,0,0,0,2,206,0,0,0,0,0,184,111,0,0,0,177,57] >;

C2×C153C8 in GAP, Magma, Sage, TeX

C_2\times C_{15}\rtimes_3C_8
% in TeX

G:=Group("C2xC15:3C8");
// GroupNames label

G:=SmallGroup(240,70);
// by ID

G=gap.SmallGroup(240,70);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,50,964,6917]);
// Polycyclic

G:=Group<a,b,c|a^2=b^15=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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