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G = D4.D27order 432 = 24·33

The non-split extension by D4 of D27 acting via D27/C27=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.D27, C4.1D54, C36.1D6, C54.7D4, C272SD16, C12.1D18, Dic542C2, C108.1C22, C27⋊C81C2, (D4×C9).1S3, (C3×D4).1D9, C3.(D4.D9), C9.(D4.S3), (D4×C27).1C2, C6.16(C9⋊D4), C2.4(C27⋊D4), C18.16(C3⋊D4), SmallGroup(432,15)

Series: Derived Chief Lower central Upper central

C1C108 — D4.D27
C1C3C9C27C54C108Dic54 — D4.D27
C27C54C108 — D4.D27
C1C2C4D4

Generators and relations for D4.D27
 G = < a,b,c,d | a4=b2=c27=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

4C2
2C22
54C4
4C6
27Q8
27C8
2C2×C6
18Dic3
4C18
27SD16
9C3⋊C8
9Dic6
2C2×C18
6Dic9
4C54
9D4.S3
3C9⋊C8
3Dic18
2Dic27
2C2×C54
3D4.D9

Smallest permutation representation of D4.D27
On 216 points
Generators in S216
(1 91 29 62)(2 92 30 63)(3 93 31 64)(4 94 32 65)(5 95 33 66)(6 96 34 67)(7 97 35 68)(8 98 36 69)(9 99 37 70)(10 100 38 71)(11 101 39 72)(12 102 40 73)(13 103 41 74)(14 104 42 75)(15 105 43 76)(16 106 44 77)(17 107 45 78)(18 108 46 79)(19 82 47 80)(20 83 48 81)(21 84 49 55)(22 85 50 56)(23 86 51 57)(24 87 52 58)(25 88 53 59)(26 89 54 60)(27 90 28 61)(109 182 155 198)(110 183 156 199)(111 184 157 200)(112 185 158 201)(113 186 159 202)(114 187 160 203)(115 188 161 204)(116 189 162 205)(117 163 136 206)(118 164 137 207)(119 165 138 208)(120 166 139 209)(121 167 140 210)(122 168 141 211)(123 169 142 212)(124 170 143 213)(125 171 144 214)(126 172 145 215)(127 173 146 216)(128 174 147 190)(129 175 148 191)(130 176 149 192)(131 177 150 193)(132 178 151 194)(133 179 152 195)(134 180 153 196)(135 181 154 197)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 61)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(109 155)(110 156)(111 157)(112 158)(113 159)(114 160)(115 161)(116 162)(117 136)(118 137)(119 138)(120 139)(121 140)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)(133 152)(134 153)(135 154)
(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)
(1 139 29 120)(2 138 30 119)(3 137 31 118)(4 136 32 117)(5 162 33 116)(6 161 34 115)(7 160 35 114)(8 159 36 113)(9 158 37 112)(10 157 38 111)(11 156 39 110)(12 155 40 109)(13 154 41 135)(14 153 42 134)(15 152 43 133)(16 151 44 132)(17 150 45 131)(18 149 46 130)(19 148 47 129)(20 147 48 128)(21 146 49 127)(22 145 50 126)(23 144 51 125)(24 143 52 124)(25 142 53 123)(26 141 54 122)(27 140 28 121)(55 216 84 173)(56 215 85 172)(57 214 86 171)(58 213 87 170)(59 212 88 169)(60 211 89 168)(61 210 90 167)(62 209 91 166)(63 208 92 165)(64 207 93 164)(65 206 94 163)(66 205 95 189)(67 204 96 188)(68 203 97 187)(69 202 98 186)(70 201 99 185)(71 200 100 184)(72 199 101 183)(73 198 102 182)(74 197 103 181)(75 196 104 180)(76 195 105 179)(77 194 106 178)(78 193 107 177)(79 192 108 176)(80 191 82 175)(81 190 83 174)

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

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

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

72 conjugacy classes

class 1 2A2B 3 4A4B6A6B6C8A8B9A9B9C 12 18A18B18C18D···18I27A···27I36A36B36C54A···54I54J···54AA108A···108I
order122344666889991218181818···1827···2736363654···5454···54108···108
size11422108244545422242224···42···24442···24···44···4

72 irreducible representations

dim111122222222222444
type+++++++++++---
imageC1C2C2C2S3D4D6SD16D9C3⋊D4D18D27C9⋊D4D54C27⋊D4D4.S3D4.D9D4.D27
kernelD4.D27C27⋊C8Dic54D4×C27D4×C9C54C36C27C3×D4C18C12D4C6C4C2C9C3C1
# reps1111111232396918139

Matrix representation of D4.D27 in GL4(𝔽433) generated by

1000
0100
00273164
00100160
,
1000
0100
00273164
00401160
,
2517000
26319500
0010
0001
,
8431100
39534900
0029129
0096142
G:=sub<GL(4,GF(433))| [1,0,0,0,0,1,0,0,0,0,273,100,0,0,164,160],[1,0,0,0,0,1,0,0,0,0,273,401,0,0,164,160],[25,263,0,0,170,195,0,0,0,0,1,0,0,0,0,1],[84,395,0,0,311,349,0,0,0,0,291,96,0,0,29,142] >;

D4.D27 in GAP, Magma, Sage, TeX

D_4.D_{27}
% in TeX

G:=Group("D4.D27");
// GroupNames label

G:=SmallGroup(432,15);
// by ID

G=gap.SmallGroup(432,15);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,254,135,58,2804,557,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^27=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D4.D27 in TeX

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