Copied to
clipboard

## G = C4×Dic15order 240 = 24·3·5

### Direct product of C4 and Dic15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C4×Dic15
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×Dic15 — C4×Dic15
 Lower central C15 — C4×Dic15
 Upper central C1 — C2×C4

Generators and relations for C4×Dic15
G = < a,b,c | a4=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 200 in 60 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×2], C10, C10 [×2], Dic3 [×4], C12 [×2], C2×C6, C15, C42, Dic5 [×4], C20 [×2], C2×C10, C2×Dic3 [×2], C2×C12, C30, C30 [×2], C2×Dic5 [×2], C2×C20, C4×Dic3, Dic15 [×4], C60 [×2], C2×C30, C4×Dic5, C2×Dic15 [×2], C2×C60, C4×Dic15
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], D5, Dic3 [×2], D6, C42, Dic5 [×2], D10, C4×S3 [×2], C2×Dic3, D15, C4×D5 [×2], C2×Dic5, C4×Dic3, Dic15 [×2], D30, C4×Dic5, C4×D15 [×2], C2×Dic15, C4×Dic15

Smallest permutation representation of C4×Dic15
Regular action on 240 points
Generators in S240
(1 177 77 119)(2 178 78 120)(3 179 79 91)(4 180 80 92)(5 151 81 93)(6 152 82 94)(7 153 83 95)(8 154 84 96)(9 155 85 97)(10 156 86 98)(11 157 87 99)(12 158 88 100)(13 159 89 101)(14 160 90 102)(15 161 61 103)(16 162 62 104)(17 163 63 105)(18 164 64 106)(19 165 65 107)(20 166 66 108)(21 167 67 109)(22 168 68 110)(23 169 69 111)(24 170 70 112)(25 171 71 113)(26 172 72 114)(27 173 73 115)(28 174 74 116)(29 175 75 117)(30 176 76 118)(31 219 184 150)(32 220 185 121)(33 221 186 122)(34 222 187 123)(35 223 188 124)(36 224 189 125)(37 225 190 126)(38 226 191 127)(39 227 192 128)(40 228 193 129)(41 229 194 130)(42 230 195 131)(43 231 196 132)(44 232 197 133)(45 233 198 134)(46 234 199 135)(47 235 200 136)(48 236 201 137)(49 237 202 138)(50 238 203 139)(51 239 204 140)(52 240 205 141)(53 211 206 142)(54 212 207 143)(55 213 208 144)(56 214 209 145)(57 215 210 146)(58 216 181 147)(59 217 182 148)(60 218 183 149)
(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)
(1 190 16 205)(2 189 17 204)(3 188 18 203)(4 187 19 202)(5 186 20 201)(6 185 21 200)(7 184 22 199)(8 183 23 198)(9 182 24 197)(10 181 25 196)(11 210 26 195)(12 209 27 194)(13 208 28 193)(14 207 29 192)(15 206 30 191)(31 68 46 83)(32 67 47 82)(33 66 48 81)(34 65 49 80)(35 64 50 79)(36 63 51 78)(37 62 52 77)(38 61 53 76)(39 90 54 75)(40 89 55 74)(41 88 56 73)(42 87 57 72)(43 86 58 71)(44 85 59 70)(45 84 60 69)(91 223 106 238)(92 222 107 237)(93 221 108 236)(94 220 109 235)(95 219 110 234)(96 218 111 233)(97 217 112 232)(98 216 113 231)(99 215 114 230)(100 214 115 229)(101 213 116 228)(102 212 117 227)(103 211 118 226)(104 240 119 225)(105 239 120 224)(121 167 136 152)(122 166 137 151)(123 165 138 180)(124 164 139 179)(125 163 140 178)(126 162 141 177)(127 161 142 176)(128 160 143 175)(129 159 144 174)(130 158 145 173)(131 157 146 172)(132 156 147 171)(133 155 148 170)(134 154 149 169)(135 153 150 168)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E ··· 4L 5A 5B 6A 6B 6C 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 4 ··· 4 5 5 6 6 6 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 15 ··· 15 2 2 2 2 2 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 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - + + - + image C1 C2 C2 C4 C4 S3 D5 Dic3 D6 Dic5 D10 C4×S3 D15 C4×D5 Dic15 D30 C4×D15 kernel C4×Dic15 C2×Dic15 C2×C60 Dic15 C60 C2×C20 C2×C12 C20 C2×C10 C12 C2×C6 C10 C2×C4 C6 C4 C22 C2 # reps 1 2 1 8 4 1 2 2 1 4 2 4 4 8 8 4 16

Matrix representation of C4×Dic15 in GL3(𝔽61) generated by

 60 0 0 0 11 0 0 0 11
,
 60 0 0 0 45 0 0 0 19
,
 50 0 0 0 0 60 0 1 0
G:=sub<GL(3,GF(61))| [60,0,0,0,11,0,0,0,11],[60,0,0,0,45,0,0,0,19],[50,0,0,0,0,1,0,60,0] >;

C4×Dic15 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_{15}
% in TeX

G:=Group("C4xDic15");
// GroupNames label

G:=SmallGroup(240,72);
// by ID

G=gap.SmallGroup(240,72);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,964,6917]);
// Polycyclic

G:=Group<a,b,c|a^4=b^30=1,c^2=b^15,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽