direct product, abelian, monomial, 2-elementary
Aliases: C22×C54, SmallGroup(216,24)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C54 |
| C1 — C22×C54 |
| C1 — C22×C54 |
Generators and relations for C22×C54
G = < a,b,c | a2=b2=c54=1, ab=ba, ac=ca, bc=cb >
Subgroups: 64, all normal (8 characteristic)
C1, C2, C3, C22, C6, C23, C9, C2×C6, C18, C22×C6, C27, C2×C18, C54, C22×C18, C2×C54, C22×C54
Quotients: C1, C2, C3, C22, C6, C23, C9, C2×C6, C18, C22×C6, C27, C2×C18, C54, C22×C18, C2×C54, C22×C54
►Regular action on 216 points
Generators in S216
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 161)(22 162)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 197)(56 198)(57 199)(58 200)(59 201)(60 202)(61 203)(62 204)(63 205)(64 206)(65 207)(66 208)(67 209)(68 210)(69 211)(70 212)(71 213)(72 214)(73 215)(74 216)(75 163)(76 164)(77 165)(78 166)(79 167)(80 168)(81 169)(82 170)(83 171)(84 172)(85 173)(86 174)(87 175)(88 176)(89 177)(90 178)(91 179)(92 180)(93 181)(94 182)(95 183)(96 184)(97 185)(98 186)(99 187)(100 188)(101 189)(102 190)(103 191)(104 192)(105 193)(106 194)(107 195)(108 196)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(109 201)(110 202)(111 203)(112 204)(113 205)(114 206)(115 207)(116 208)(117 209)(118 210)(119 211)(120 212)(121 213)(122 214)(123 215)(124 216)(125 163)(126 164)(127 165)(128 166)(129 167)(130 168)(131 169)(132 170)(133 171)(134 172)(135 173)(136 174)(137 175)(138 176)(139 177)(140 178)(141 179)(142 180)(143 181)(144 182)(145 183)(146 184)(147 185)(148 186)(149 187)(150 188)(151 189)(152 190)(153 191)(154 192)(155 193)(156 194)(157 195)(158 196)(159 197)(160 198)(161 199)(162 200)
(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)
G:=sub<Sym(216)| (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,197)(56,198)(57,199)(58,200)(59,201)(60,202)(61,203)(62,204)(63,205)(64,206)(65,207)(66,208)(67,209)(68,210)(69,211)(70,212)(71,213)(72,214)(73,215)(74,216)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176)(89,177)(90,178)(91,179)(92,180)(93,181)(94,182)(95,183)(96,184)(97,185)(98,186)(99,187)(100,188)(101,189)(102,190)(103,191)(104,192)(105,193)(106,194)(107,195)(108,196), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(109,201)(110,202)(111,203)(112,204)(113,205)(114,206)(115,207)(116,208)(117,209)(118,210)(119,211)(120,212)(121,213)(122,214)(123,215)(124,216)(125,163)(126,164)(127,165)(128,166)(129,167)(130,168)(131,169)(132,170)(133,171)(134,172)(135,173)(136,174)(137,175)(138,176)(139,177)(140,178)(141,179)(142,180)(143,181)(144,182)(145,183)(146,184)(147,185)(148,186)(149,187)(150,188)(151,189)(152,190)(153,191)(154,192)(155,193)(156,194)(157,195)(158,196)(159,197)(160,198)(161,199)(162,200), (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)>;
G:=Group( (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,197)(56,198)(57,199)(58,200)(59,201)(60,202)(61,203)(62,204)(63,205)(64,206)(65,207)(66,208)(67,209)(68,210)(69,211)(70,212)(71,213)(72,214)(73,215)(74,216)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176)(89,177)(90,178)(91,179)(92,180)(93,181)(94,182)(95,183)(96,184)(97,185)(98,186)(99,187)(100,188)(101,189)(102,190)(103,191)(104,192)(105,193)(106,194)(107,195)(108,196), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(109,201)(110,202)(111,203)(112,204)(113,205)(114,206)(115,207)(116,208)(117,209)(118,210)(119,211)(120,212)(121,213)(122,214)(123,215)(124,216)(125,163)(126,164)(127,165)(128,166)(129,167)(130,168)(131,169)(132,170)(133,171)(134,172)(135,173)(136,174)(137,175)(138,176)(139,177)(140,178)(141,179)(142,180)(143,181)(144,182)(145,183)(146,184)(147,185)(148,186)(149,187)(150,188)(151,189)(152,190)(153,191)(154,192)(155,193)(156,194)(157,195)(158,196)(159,197)(160,198)(161,199)(162,200), (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) );
G=PermutationGroup([[(1,141),(2,142),(3,143),(4,144),(5,145),(6,146),(7,147),(8,148),(9,149),(10,150),(11,151),(12,152),(13,153),(14,154),(15,155),(16,156),(17,157),(18,158),(19,159),(20,160),(21,161),(22,162),(23,109),(24,110),(25,111),(26,112),(27,113),(28,114),(29,115),(30,116),(31,117),(32,118),(33,119),(34,120),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,127),(42,128),(43,129),(44,130),(45,131),(46,132),(47,133),(48,134),(49,135),(50,136),(51,137),(52,138),(53,139),(54,140),(55,197),(56,198),(57,199),(58,200),(59,201),(60,202),(61,203),(62,204),(63,205),(64,206),(65,207),(66,208),(67,209),(68,210),(69,211),(70,212),(71,213),(72,214),(73,215),(74,216),(75,163),(76,164),(77,165),(78,166),(79,167),(80,168),(81,169),(82,170),(83,171),(84,172),(85,173),(86,174),(87,175),(88,176),(89,177),(90,178),(91,179),(92,180),(93,181),(94,182),(95,183),(96,184),(97,185),(98,186),(99,187),(100,188),(101,189),(102,190),(103,191),(104,192),(105,193),(106,194),(107,195),(108,196)], [(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,97),(8,98),(9,99),(10,100),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(109,201),(110,202),(111,203),(112,204),(113,205),(114,206),(115,207),(116,208),(117,209),(118,210),(119,211),(120,212),(121,213),(122,214),(123,215),(124,216),(125,163),(126,164),(127,165),(128,166),(129,167),(130,168),(131,169),(132,170),(133,171),(134,172),(135,173),(136,174),(137,175),(138,176),(139,177),(140,178),(141,179),(142,180),(143,181),(144,182),(145,183),(146,184),(147,185),(148,186),(149,187),(150,188),(151,189),(152,190),(153,191),(154,192),(155,193),(156,194),(157,195),(158,196),(159,197),(160,198),(161,199),(162,200)], [(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)]])
C22×C54 is a maximal subgroup of
C54.D4
216 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 6A | ··· | 6N | 9A | ··· | 9F | 18A | ··· | 18AP | 27A | ··· | 27R | 54A | ··· | 54DV |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 |
| kernel | C22×C54 | C2×C54 | C22×C18 | C2×C18 | C22×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 7 | 2 | 14 | 6 | 42 | 18 | 126 |
Matrix representation of C22×C54 ►in GL3(𝔽109) generated by
| 1 | 0 | 0 |
| 0 | 108 | 0 |
| 0 | 0 | 1 |
| 108 | 0 | 0 |
| 0 | 108 | 0 |
| 0 | 0 | 108 |
| 36 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 9 |
G:=sub<GL(3,GF(109))| [1,0,0,0,108,0,0,0,1],[108,0,0,0,108,0,0,0,108],[36,0,0,0,22,0,0,0,9] >;
C22×C54 in GAP, Magma, Sage, TeX
C_2^2\times C_{54} % in TeX
G:=Group("C2^2xC54"); // GroupNames label
G:=SmallGroup(216,24);
// by ID
G=gap.SmallGroup(216,24);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,93,118]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^54=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations