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G = C8×D27order 432 = 24·33

Direct product of C8 and D27

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C8×D27, C2163C2, C72.6S3, C24.7D9, D54.2C4, C4.13D54, C36.60D6, C12.60D18, Dic27.2C4, C108.13C22, C9.(S3×C8), C27⋊C86C2, C3.(C8×D9), C271(C2×C8), C6.5(C4×D9), C18.6(C4×S3), C54.1(C2×C4), C2.1(C4×D27), (C4×D27).3C2, SmallGroup(432,5)

Series: Derived Chief Lower central Upper central

C1C27 — C8×D27
C1C3C9C27C54C108C4×D27 — C8×D27
C27 — C8×D27
C1C8

Generators and relations for C8×D27
 G = < a,b,c | a8=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >

27C2
27C2
27C22
27C4
9S3
9S3
27C8
27C2×C4
9Dic3
9D6
3D9
3D9
27C2×C8
9C4×S3
9C3⋊C8
3D18
3Dic9
9S3×C8
3C9⋊C8
3C4×D9
3C8×D9

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

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

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

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

120 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 8A8B8C8D8E8F8G8H9A9B9C12A12B18A18B18C24A24B24C24D27A···27I36A···36F54A···54I72A···72L108A···108R216A···216AJ
order12223444468888888899912121818182424242427···2736···3654···5472···72108···108216···216
size112727211272721111272727272222222222222···22···22···22···22···22···2

120 irreducible representations

dim1111111222222222222
type++++++++++
imageC1C2C2C2C4C4C8S3D6D9C4×S3D18S3×C8D27C4×D9D54C8×D9C4×D27C8×D27
kernelC8×D27C27⋊C8C216C4×D27Dic27D54D27C72C36C24C18C12C9C8C6C4C3C2C1
# reps1111228113234969121836

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

148000
014800
001790
000179
,
10100
32242200
00271130
00303401
,
42243200
1201100
0001
0010
G:=sub<GL(4,GF(433))| [148,0,0,0,0,148,0,0,0,0,179,0,0,0,0,179],[10,322,0,0,1,422,0,0,0,0,271,303,0,0,130,401],[422,120,0,0,432,11,0,0,0,0,0,1,0,0,1,0] >;

C8×D27 in GAP, Magma, Sage, TeX

C_8\times D_{27}
% in TeX

G:=Group("C8xD27");
// GroupNames label

G:=SmallGroup(432,5);
// by ID

G=gap.SmallGroup(432,5);
# by ID

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

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

Export

Subgroup lattice of C8×D27 in TeX

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