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G = C8⋊D27order 432 = 24·33

3rd semidirect product of C8 and D27 acting via D27/C27=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83D27, D54.C4, C2164C2, C72.7S3, C24.8D9, C4.14D54, C36.61D6, Dic27.C4, C271M4(2), C12.61D18, C108.14C22, C27⋊C84C2, C6.6(C4×D9), C9.(C8⋊S3), C18.7(C4×S3), C54.2(C2×C4), C3.(C8⋊D9), C2.3(C4×D27), (C4×D27).2C2, SmallGroup(432,6)

Series: Derived Chief Lower central Upper central

C1C54 — C8⋊D27
C1C3C9C27C54C108C4×D27 — C8⋊D27
C27C54 — C8⋊D27
C1C4C8

Generators and relations for C8⋊D27
 G = < a,b,c | a8=b27=c2=1, ab=ba, cac=a5, cbc=b-1 >

54C2
27C22
27C4
18S3
27C8
27C2×C4
9D6
9Dic3
6D9
27M4(2)
9C4×S3
9C3⋊C8
3D18
3Dic9
2D27
9C8⋊S3
3C4×D9
3C9⋊C8
3C8⋊D9

Smallest permutation representation of C8⋊D27
On 216 points
Generators in S216
(1 215 83 146 53 181 71 111)(2 216 84 147 54 182 72 112)(3 190 85 148 28 183 73 113)(4 191 86 149 29 184 74 114)(5 192 87 150 30 185 75 115)(6 193 88 151 31 186 76 116)(7 194 89 152 32 187 77 117)(8 195 90 153 33 188 78 118)(9 196 91 154 34 189 79 119)(10 197 92 155 35 163 80 120)(11 198 93 156 36 164 81 121)(12 199 94 157 37 165 55 122)(13 200 95 158 38 166 56 123)(14 201 96 159 39 167 57 124)(15 202 97 160 40 168 58 125)(16 203 98 161 41 169 59 126)(17 204 99 162 42 170 60 127)(18 205 100 136 43 171 61 128)(19 206 101 137 44 172 62 129)(20 207 102 138 45 173 63 130)(21 208 103 139 46 174 64 131)(22 209 104 140 47 175 65 132)(23 210 105 141 48 176 66 133)(24 211 106 142 49 177 67 134)(25 212 107 143 50 178 68 135)(26 213 108 144 51 179 69 109)(27 214 82 145 52 180 70 110)
(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 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(52 54)(55 60)(56 59)(57 58)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(82 84)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(109 148)(110 147)(111 146)(112 145)(113 144)(114 143)(115 142)(116 141)(117 140)(118 139)(119 138)(120 137)(121 136)(122 162)(123 161)(124 160)(125 159)(126 158)(127 157)(128 156)(129 155)(130 154)(131 153)(132 152)(133 151)(134 150)(135 149)(163 206)(164 205)(165 204)(166 203)(167 202)(168 201)(169 200)(170 199)(171 198)(172 197)(173 196)(174 195)(175 194)(176 193)(177 192)(178 191)(179 190)(180 216)(181 215)(182 214)(183 213)(184 212)(185 211)(186 210)(187 209)(188 208)(189 207)

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

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

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

114 conjugacy classes

class 1 2A2B 3 4A4B4C 6 8A8B8C8D9A9B9C12A12B18A18B18C24A24B24C24D27A···27I36A···36F54A···54I72A···72L108A···108R216A···216AJ
order12234446888899912121818182424242427···2736···3654···5472···72108···108216···216
size11542115422254542222222222222···22···22···22···22···22···2

114 irreducible representations

dim1111112222222222222
type++++++++++
imageC1C2C2C2C4C4S3D6M4(2)D9C4×S3D18C8⋊S3D27C4×D9D54C8⋊D9C4×D27C8⋊D27
kernelC8⋊D27C27⋊C8C216C4×D27Dic27D54C72C36C27C24C18C12C9C8C6C4C3C2C1
# reps1111221123234969121836

Matrix representation of C8⋊D27 in GL4(𝔽433) generated by

179000
017900
0036169
00145397
,
2632500
40823800
00315184
00122129
,
1000
43243200
00118249
00351315
G:=sub<GL(4,GF(433))| [179,0,0,0,0,179,0,0,0,0,36,145,0,0,169,397],[263,408,0,0,25,238,0,0,0,0,315,122,0,0,184,129],[1,432,0,0,0,432,0,0,0,0,118,351,0,0,249,315] >;

C8⋊D27 in GAP, Magma, Sage, TeX

C_8\rtimes D_{27}
% in TeX

G:=Group("C8:D27");
// GroupNames label

G:=SmallGroup(432,6);
// by ID

G=gap.SmallGroup(432,6);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,58,2804,557,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^8=b^27=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊D27 in TeX

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