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G = D4⋊D27order 432 = 24·33

The semidirect product of D4 and D27 acting via D27/C27=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4⋊D27, C272D8, C4.2D54, C54.8D4, C36.2D6, D1082C2, C12.2D18, C108.2C22, C27⋊C82C2, C3.(D4⋊D9), C9.(D4⋊S3), (D4×C27)⋊1C2, (C3×D4).2D9, (D4×C9).2S3, C6.17(C9⋊D4), C2.5(C27⋊D4), C18.17(C3⋊D4), SmallGroup(432,16)

Series: Derived Chief Lower central Upper central

C1C108 — D4⋊D27
C1C3C9C27C54C108D108 — D4⋊D27
C27C54C108 — D4⋊D27
C1C2C4D4

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

4C2
108C2
2C22
54C22
4C6
36S3
27D4
27C8
2C2×C6
18D6
4C18
12D9
27D8
9C3⋊C8
9D12
2C2×C18
6D18
4C54
4D27
9D4⋊S3
3D36
3C9⋊C8
2D54
2C2×C54
3D4⋊D9

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

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

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

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

72 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C8A8B9A9B9C 12 18A18B18C18D···18I27A···27I36A36B36C54A···54I54J···54AA108A···108I
order122234666889991218181818···1827···2736363654···5454···54108···108
size11410822244545422242224···42···24442···24···44···4

72 irreducible representations

dim111122222222222444
type+++++++++++++++
imageC1C2C2C2S3D4D6D8D9C3⋊D4D18D27C9⋊D4D54C27⋊D4D4⋊S3D4⋊D9D4⋊D27
kernelD4⋊D27C27⋊C8D108D4×C27D4×C9C54C36C27C3×D4C18C12D4C6C4C2C9C3C1
# reps1111111232396918139

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

1000
0100
0001
004320
,
432000
043200
00330103
00103103
,
40113000
30327100
0010
0001
,
2519500
17040800
0010
000432
G:=sub<GL(4,GF(433))| [1,0,0,0,0,1,0,0,0,0,0,432,0,0,1,0],[432,0,0,0,0,432,0,0,0,0,330,103,0,0,103,103],[401,303,0,0,130,271,0,0,0,0,1,0,0,0,0,1],[25,170,0,0,195,408,0,0,0,0,1,0,0,0,0,432] >;

D4⋊D27 in GAP, Magma, Sage, TeX

D_4\rtimes D_{27}
% in TeX

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

G:=SmallGroup(432,16);
// by ID

G=gap.SmallGroup(432,16);
# by ID

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

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

Export

Subgroup lattice of D4⋊D27 in TeX

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