Copied to
clipboard

G = M4(2)×C27order 432 = 24·33

Direct product of C27 and M4(2)

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: M4(2)×C27, C83C54, C4.C108, C2167C2, C72.11C6, C108.4C4, C24.7C18, C12.5C36, C22.C108, C36.11C12, C108.22C22, (C2×C54).1C4, C4.5(C2×C54), (C2×C6).3C36, (C2×C4).2C54, C3.(C9×M4(2)), C9.(C3×M4(2)), C2.3(C2×C108), C36.51(C2×C6), (C2×C18).7C12, (C2×C108).8C2, (C2×C36).21C6, C6.12(C2×C36), (C2×C12).9C18, C54.12(C2×C4), (C9×M4(2)).C3, (C3×M4(2)).C9, C18.26(C2×C12), C12.28(C2×C18), SmallGroup(432,24)

Series: Derived Chief Lower central Upper central

C1C2 — M4(2)×C27
C1C3C6C18C36C108C216 — M4(2)×C27
C1C2 — M4(2)×C27
C1C108 — M4(2)×C27

Generators and relations for M4(2)×C27
 G = < a,b,c | a27=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

2C2
2C6
2C18
2C54

Smallest permutation representation of M4(2)×C27
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)
(1 138 170 113 40 106 199 66)(2 139 171 114 41 107 200 67)(3 140 172 115 42 108 201 68)(4 141 173 116 43 82 202 69)(5 142 174 117 44 83 203 70)(6 143 175 118 45 84 204 71)(7 144 176 119 46 85 205 72)(8 145 177 120 47 86 206 73)(9 146 178 121 48 87 207 74)(10 147 179 122 49 88 208 75)(11 148 180 123 50 89 209 76)(12 149 181 124 51 90 210 77)(13 150 182 125 52 91 211 78)(14 151 183 126 53 92 212 79)(15 152 184 127 54 93 213 80)(16 153 185 128 28 94 214 81)(17 154 186 129 29 95 215 55)(18 155 187 130 30 96 216 56)(19 156 188 131 31 97 190 57)(20 157 189 132 32 98 191 58)(21 158 163 133 33 99 192 59)(22 159 164 134 34 100 193 60)(23 160 165 135 35 101 194 61)(24 161 166 109 36 102 195 62)(25 162 167 110 37 103 196 63)(26 136 168 111 38 104 197 64)(27 137 169 112 39 105 198 65)
(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 135)(62 109)(63 110)(64 111)(65 112)(66 113)(67 114)(68 115)(69 116)(70 117)(71 118)(72 119)(73 120)(74 121)(75 122)(76 123)(77 124)(78 125)(79 126)(80 127)(81 128)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 136)(105 137)(106 138)(107 139)(108 140)

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

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

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

270 conjugacy classes

class 1 2A2B3A3B4A4B4C6A6B6C6D8A8B8C8D9A···9F12A12B12C12D12E12F18A···18F18G···18L24A···24H27A···27R36A···36L36M···36R54A···54R54S···54AJ72A···72X108A···108AJ108AK···108BB216A···216BT
order12233444666688889···912121212121218···1818···1824···2427···2736···3636···3654···5454···5472···72108···108108···108216···216
size11211112112222221···11111221···12···22···21···11···12···21···12···22···21···12···22···2

270 irreducible representations

dim111111111111111111112222
type+++
imageC1C2C2C3C4C4C6C6C9C12C12C18C18C27C36C36C54C54C108C108M4(2)C3×M4(2)C9×M4(2)M4(2)×C27
kernelM4(2)×C27C216C2×C108C9×M4(2)C108C2×C54C72C2×C36C3×M4(2)C36C2×C18C24C2×C12M4(2)C12C2×C6C8C2×C4C4C22C27C9C3C1
# reps1212224264412618121236183636241236

Matrix representation of M4(2)×C27 in GL2(𝔽433) generated by

2690
0269
,
254431
343179
,
10
254432
G:=sub<GL(2,GF(433))| [269,0,0,269],[254,343,431,179],[1,254,0,432] >;

M4(2)×C27 in GAP, Magma, Sage, TeX

M_4(2)\times C_{27}
% in TeX

G:=Group("M4(2)xC27");
// GroupNames label

G:=SmallGroup(432,24);
// by ID

G=gap.SmallGroup(432,24);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,84,1037,142,192,242]);
// Polycyclic

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

Export

Subgroup lattice of M4(2)×C27 in TeX

׿
×
𝔽