Copied to
clipboard

## G = C2×C15⋊C8order 240 = 24·3·5

### Direct product of C2 and C15⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×C15⋊C8
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C15⋊C8 — C2×C15⋊C8
 Lower central C15 — C2×C15⋊C8
 Upper central C1 — C22

Generators and relations for C2×C15⋊C8
G = < a,b,c | a2=b15=c8=1, ab=ba, ac=ca, cbc-1=b2 >

Smallest permutation representation of C2×C15⋊C8
Regular action on 240 points
Generators in S240
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 71)(17 72)(18 73)(19 74)(20 75)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 106)(45 107)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 105)(121 196)(122 197)(123 198)(124 199)(125 200)(126 201)(127 202)(128 203)(129 204)(130 205)(131 206)(132 207)(133 208)(134 209)(135 210)(136 186)(137 187)(138 188)(139 189)(140 190)(141 191)(142 192)(143 193)(144 194)(145 195)(146 181)(147 182)(148 183)(149 184)(150 185)(151 240)(152 226)(153 227)(154 228)(155 229)(156 230)(157 231)(158 232)(159 233)(160 234)(161 235)(162 236)(163 237)(164 238)(165 239)(166 212)(167 213)(168 214)(169 215)(170 216)(171 217)(172 218)(173 219)(174 220)(175 221)(176 222)(177 223)(178 224)(179 225)(180 211)
(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)
(1 232 49 196 24 211 35 181)(2 240 53 198 25 219 39 183)(3 233 57 200 26 212 43 185)(4 226 46 202 27 220 32 187)(5 234 50 204 28 213 36 189)(6 227 54 206 29 221 40 191)(7 235 58 208 30 214 44 193)(8 228 47 210 16 222 33 195)(9 236 51 197 17 215 37 182)(10 229 55 199 18 223 41 184)(11 237 59 201 19 216 45 186)(12 230 48 203 20 224 34 188)(13 238 52 205 21 217 38 190)(14 231 56 207 22 225 42 192)(15 239 60 209 23 218 31 194)(61 171 115 140 81 164 97 130)(62 179 119 142 82 157 101 132)(63 172 108 144 83 165 105 134)(64 180 112 146 84 158 94 121)(65 173 116 148 85 151 98 123)(66 166 120 150 86 159 102 125)(67 174 109 137 87 152 91 127)(68 167 113 139 88 160 95 129)(69 175 117 141 89 153 99 131)(70 168 106 143 90 161 103 133)(71 176 110 145 76 154 92 135)(72 169 114 147 77 162 96 122)(73 177 118 149 78 155 100 124)(74 170 107 136 79 163 104 126)(75 178 111 138 80 156 93 128)

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

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

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

C2×C15⋊C8 is a maximal subgroup of
Dic3×C5⋊C8  C30.M4(2)  Dic5.22D12  D30⋊C8  C30.4M4(2)  Dic15⋊C8  C60⋊C8  C30.11C42  C30.7M4(2)  Dic5.13D12  C30.22M4(2)  C2×S3×C5⋊C8  D15⋊C8⋊C2  Dic10.Dic3
C2×C15⋊C8 is a maximal quotient of
C60.C8  C60⋊C8  C30.22M4(2)

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + - + - + - + image C1 C2 C2 C4 C4 C8 S3 Dic3 D6 Dic3 C3⋊C8 F5 C5⋊C8 C2×F5 C3⋊F5 C15⋊C8 C2×C3⋊F5 kernel C2×C15⋊C8 C15⋊C8 C6×Dic5 C3×Dic5 C2×C30 C30 C2×Dic5 Dic5 Dic5 C2×C10 C10 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 2 8 1 1 1 1 4 1 2 1 2 4 2

Matrix representation of C2×C15⋊C8 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 240 0 0 0 0 1 240 0 0 0 0 0 0 154 36 142 0 0 0 0 0 240 0 0 0 1 0 240 0 0 0 226 45 16 87
,
 191 106 0 0 0 0 56 50 0 0 0 0 0 0 98 61 63 61 0 0 187 189 28 34 0 0 127 217 78 42 0 0 38 42 130 117

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,240,240,0,0,0,0,0,0,154,0,1,226,0,0,36,0,0,45,0,0,142,240,240,16,0,0,0,0,0,87],[191,56,0,0,0,0,106,50,0,0,0,0,0,0,98,187,127,38,0,0,61,189,217,42,0,0,63,28,78,130,0,0,61,34,42,117] >;

C2×C15⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_{15}\rtimes C_8
% in TeX

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

G:=SmallGroup(240,122);
// by ID

G=gap.SmallGroup(240,122);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,50,964,5189,1745]);
// 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^2>;
// generators/relations

Export

׿
×
𝔽