metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊2D27, C216⋊2C2, C72.4S3, C54.2D4, C24.4D9, C4.9D54, C6.2D36, C27⋊1SD16, C18.2D12, C2.4D108, C36.52D6, Dic54⋊1C2, D108.1C2, C12.52D18, C108.9C22, C9.(C24⋊C2), C3.(C72⋊C2), SmallGroup(432,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C216⋊C2
G = < a,b | a216=b2=1, bab=a107 >
Smallest permutation representation of C216⋊C2
►On 216 points
Generators in S216
(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)
(2 108)(3 215)(4 106)(5 213)(6 104)(7 211)(8 102)(9 209)(10 100)(11 207)(12 98)(13 205)(14 96)(15 203)(16 94)(17 201)(18 92)(19 199)(20 90)(21 197)(22 88)(23 195)(24 86)(25 193)(26 84)(27 191)(28 82)(29 189)(30 80)(31 187)(32 78)(33 185)(34 76)(35 183)(36 74)(37 181)(38 72)(39 179)(40 70)(41 177)(42 68)(43 175)(44 66)(45 173)(46 64)(47 171)(48 62)(49 169)(50 60)(51 167)(52 58)(53 165)(54 56)(55 163)(57 161)(59 159)(61 157)(63 155)(65 153)(67 151)(69 149)(71 147)(73 145)(75 143)(77 141)(79 139)(81 137)(83 135)(85 133)(87 131)(89 129)(91 127)(93 125)(95 123)(97 121)(99 119)(101 117)(103 115)(105 113)(107 111)(110 216)(112 214)(114 212)(116 210)(118 208)(120 206)(122 204)(124 202)(126 200)(128 198)(130 196)(132 194)(134 192)(136 190)(138 188)(140 186)(142 184)(144 182)(146 180)(148 178)(150 176)(152 174)(154 172)(156 170)(158 168)(160 166)(162 164)
G:=sub<Sym(216)| (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), (2,108)(3,215)(4,106)(5,213)(6,104)(7,211)(8,102)(9,209)(10,100)(11,207)(12,98)(13,205)(14,96)(15,203)(16,94)(17,201)(18,92)(19,199)(20,90)(21,197)(22,88)(23,195)(24,86)(25,193)(26,84)(27,191)(28,82)(29,189)(30,80)(31,187)(32,78)(33,185)(34,76)(35,183)(36,74)(37,181)(38,72)(39,179)(40,70)(41,177)(42,68)(43,175)(44,66)(45,173)(46,64)(47,171)(48,62)(49,169)(50,60)(51,167)(52,58)(53,165)(54,56)(55,163)(57,161)(59,159)(61,157)(63,155)(65,153)(67,151)(69,149)(71,147)(73,145)(75,143)(77,141)(79,139)(81,137)(83,135)(85,133)(87,131)(89,129)(91,127)(93,125)(95,123)(97,121)(99,119)(101,117)(103,115)(105,113)(107,111)(110,216)(112,214)(114,212)(116,210)(118,208)(120,206)(122,204)(124,202)(126,200)(128,198)(130,196)(132,194)(134,192)(136,190)(138,188)(140,186)(142,184)(144,182)(146,180)(148,178)(150,176)(152,174)(154,172)(156,170)(158,168)(160,166)(162,164)>;
G:=Group( (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), (2,108)(3,215)(4,106)(5,213)(6,104)(7,211)(8,102)(9,209)(10,100)(11,207)(12,98)(13,205)(14,96)(15,203)(16,94)(17,201)(18,92)(19,199)(20,90)(21,197)(22,88)(23,195)(24,86)(25,193)(26,84)(27,191)(28,82)(29,189)(30,80)(31,187)(32,78)(33,185)(34,76)(35,183)(36,74)(37,181)(38,72)(39,179)(40,70)(41,177)(42,68)(43,175)(44,66)(45,173)(46,64)(47,171)(48,62)(49,169)(50,60)(51,167)(52,58)(53,165)(54,56)(55,163)(57,161)(59,159)(61,157)(63,155)(65,153)(67,151)(69,149)(71,147)(73,145)(75,143)(77,141)(79,139)(81,137)(83,135)(85,133)(87,131)(89,129)(91,127)(93,125)(95,123)(97,121)(99,119)(101,117)(103,115)(105,113)(107,111)(110,216)(112,214)(114,212)(116,210)(118,208)(120,206)(122,204)(124,202)(126,200)(128,198)(130,196)(132,194)(134,192)(136,190)(138,188)(140,186)(142,184)(144,182)(146,180)(148,178)(150,176)(152,174)(154,172)(156,170)(158,168)(160,166)(162,164) );
G=PermutationGroup([[(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)], [(2,108),(3,215),(4,106),(5,213),(6,104),(7,211),(8,102),(9,209),(10,100),(11,207),(12,98),(13,205),(14,96),(15,203),(16,94),(17,201),(18,92),(19,199),(20,90),(21,197),(22,88),(23,195),(24,86),(25,193),(26,84),(27,191),(28,82),(29,189),(30,80),(31,187),(32,78),(33,185),(34,76),(35,183),(36,74),(37,181),(38,72),(39,179),(40,70),(41,177),(42,68),(43,175),(44,66),(45,173),(46,64),(47,171),(48,62),(49,169),(50,60),(51,167),(52,58),(53,165),(54,56),(55,163),(57,161),(59,159),(61,157),(63,155),(65,153),(67,151),(69,149),(71,147),(73,145),(75,143),(77,141),(79,139),(81,137),(83,135),(85,133),(87,131),(89,129),(91,127),(93,125),(95,123),(97,121),(99,119),(101,117),(103,115),(105,113),(107,111),(110,216),(112,214),(114,212),(116,210),(118,208),(120,206),(122,204),(124,202),(126,200),(128,198),(130,196),(132,194),(134,192),(136,190),(138,188),(140,186),(142,184),(144,182),(146,180),(148,178),(150,176),(152,174),(154,172),(156,170),(158,168),(160,166),(162,164)]])
111 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 27A | ··· | 27I | 36A | ··· | 36F | 54A | ··· | 54I | 72A | ··· | 72L | 108A | ··· | 108R | 216A | ··· | 216AJ |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 72 | ··· | 72 | 108 | ··· | 108 | 216 | ··· | 216 |
| size | 1 | 1 | 108 | 2 | 2 | 108 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
111 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | SD16 | D9 | D12 | D18 | C24⋊C2 | D27 | D36 | D54 | C72⋊C2 | D108 | C216⋊C2 |
| kernel | C216⋊C2 | C216 | Dic54 | D108 | C72 | C54 | C36 | C27 | C24 | C18 | C12 | C9 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 4 | 9 | 6 | 9 | 12 | 18 | 36 |
Matrix representation of C216⋊C2 ►in GL2(𝔽433) generated by
| 15 | 57 |
| 376 | 72 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(433))| [15,376,57,72],[0,1,1,0] >;
C216⋊C2 in GAP, Magma, Sage, TeX
C_{216}\rtimes C_2 % in TeX
G:=Group("C216:C2"); // GroupNames label
G:=SmallGroup(432,7);
// by ID
G=gap.SmallGroup(432,7);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,36,254,58,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b|a^216=b^2=1,b*a*b=a^107>;
// generators/relations
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