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G = C27⋊C8order 216 = 23·33

The semidirect product of C27 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C27⋊C8, C54.C4, C36.4S3, C4.2D27, C12.4D9, C2.Dic27, C108.2C2, C6.1Dic9, C18.1Dic3, C9.(C3⋊C8), C3.(C9⋊C8), SmallGroup(216,1)

Series: Derived Chief Lower central Upper central

C1C27 — C27⋊C8
C1C3C9C27C54C108 — C27⋊C8
C27 — C27⋊C8
C1C4

Generators and relations for C27⋊C8
 G = < a,b | a27=b8=1, bab-1=a-1 >

27C8
9C3⋊C8
3C9⋊C8

Smallest permutation representation of C27⋊C8
Regular action 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)
(1 205 88 139 37 185 64 116)(2 204 89 138 38 184 65 115)(3 203 90 137 39 183 66 114)(4 202 91 136 40 182 67 113)(5 201 92 162 41 181 68 112)(6 200 93 161 42 180 69 111)(7 199 94 160 43 179 70 110)(8 198 95 159 44 178 71 109)(9 197 96 158 45 177 72 135)(10 196 97 157 46 176 73 134)(11 195 98 156 47 175 74 133)(12 194 99 155 48 174 75 132)(13 193 100 154 49 173 76 131)(14 192 101 153 50 172 77 130)(15 191 102 152 51 171 78 129)(16 190 103 151 52 170 79 128)(17 216 104 150 53 169 80 127)(18 215 105 149 54 168 81 126)(19 214 106 148 28 167 55 125)(20 213 107 147 29 166 56 124)(21 212 108 146 30 165 57 123)(22 211 82 145 31 164 58 122)(23 210 83 144 32 163 59 121)(24 209 84 143 33 189 60 120)(25 208 85 142 34 188 61 119)(26 207 86 141 35 187 62 118)(27 206 87 140 36 186 63 117)

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

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

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

C27⋊C8 is a maximal subgroup of   C8×D27  C8⋊D27  C4.Dic27  D4.D27  D4⋊D27  C27⋊Q16  Q82D27
C27⋊C8 is a maximal quotient of   C27⋊C16

60 conjugacy classes

class 1  2  3 4A4B 6 8A8B8C8D9A9B9C12A12B18A18B18C27A···27I36A···36F54A···54I108A···108R
order1234468888999121218181827···2736···3654···54108···108
size11211227272727222222222···22···22···22···2

60 irreducible representations

dim1111222222222
type+++-+-+-
imageC1C2C4C8S3Dic3D9C3⋊C8Dic9D27C9⋊C8Dic27C27⋊C8
kernelC27⋊C8C108C54C27C36C18C12C9C6C4C3C2C1
# reps11241132396918

Matrix representation of C27⋊C8 in GL2(𝔽433) generated by

401130
303271
,
91298
207342
G:=sub<GL(2,GF(433))| [401,303,130,271],[91,207,298,342] >;

C27⋊C8 in GAP, Magma, Sage, TeX

C_{27}\rtimes C_8
% in TeX

G:=Group("C27:C8");
// GroupNames label

G:=SmallGroup(216,1);
// by ID

G=gap.SmallGroup(216,1);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,12,31,963,381,3604,208,5189]);
// Polycyclic

G:=Group<a,b|a^27=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C27⋊C8 in TeX

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