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G = C216⋊C2order 432 = 24·33

2nd semidirect product of C216 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C82D27, C2162C2, C72.4S3, C54.2D4, C24.4D9, C4.9D54, C6.2D36, C271SD16, C18.2D12, C2.4D108, C36.52D6, Dic541C2, D108.1C2, C12.52D18, C108.9C22, C9.(C24⋊C2), C3.(C72⋊C2), SmallGroup(432,7)

Series: Derived Chief Lower central Upper central

C1C108 — C216⋊C2
C1C3C9C27C54C108D108 — C216⋊C2
C27C54C108 — C216⋊C2
C1C2C4C8

Generators and relations for C216⋊C2
 G = < a,b | a216=b2=1, bab=a107 >

108C2
54C22
54C4
36S3
27Q8
27D4
18D6
18Dic3
12D9
27SD16
9D12
9Dic6
6Dic9
6D18
4D27
9C24⋊C2
3Dic18
3D36
2Dic27
2D54
3C72⋊C2

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 2A2B 3 4A4B 6 8A8B9A9B9C12A12B18A18B18C24A24B24C24D27A···27I36A···36F54A···54I72A···72L108A···108R216A···216AJ
order12234468899912121818182424242427···2736···3654···5472···72108···108216···216
size11108221082222222222222222···22···22···22···22···22···2

111 irreducible representations

dim111122222222222222
type++++++++++++++
imageC1C2C2C2S3D4D6SD16D9D12D18C24⋊C2D27D36D54C72⋊C2D108C216⋊C2
kernelC216⋊C2C216Dic54D108C72C54C36C27C24C18C12C9C8C6C4C3C2C1
# reps111111123234969121836

Matrix representation of C216⋊C2 in GL2(𝔽433) generated by

1557
37672
,
01
10
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

Export

Subgroup lattice of C216⋊C2 in TeX

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