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## G = D4×D27order 432 = 24·33

### Direct product of D4 and D27

Series: Derived Chief Lower central Upper central

 Derived series C1 — C54 — D4×D27
 Chief series C1 — C3 — C9 — C27 — C54 — D54 — C22×D27 — D4×D27
 Lower central C27 — C54 — D4×D27
 Upper central C1 — C2 — D4

Generators and relations for D4×D27
G = < a,b,c,d | a4=b2=c27=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1040 in 108 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, C23, C9, Dic3, C12, D6, C2×C6, C2×D4, D9, C18, C18, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C27, Dic9, C36, D18, C2×C18, S3×D4, D27, D27, C54, C54, C4×D9, D36, C9⋊D4, D4×C9, C22×D9, Dic27, C108, D54, D54, D54, C2×C54, D4×D9, C4×D27, D108, C27⋊D4, D4×C27, C22×D27, D4×D27
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, D18, S3×D4, D27, C22×D9, D54, D4×D9, C22×D27, D4×D27

Smallest permutation representation of D4×D27
On 108 points
Generators in S108
(1 94 29 65)(2 95 30 66)(3 96 31 67)(4 97 32 68)(5 98 33 69)(6 99 34 70)(7 100 35 71)(8 101 36 72)(9 102 37 73)(10 103 38 74)(11 104 39 75)(12 105 40 76)(13 106 41 77)(14 107 42 78)(15 108 43 79)(16 82 44 80)(17 83 45 81)(18 84 46 55)(19 85 47 56)(20 86 48 57)(21 87 49 58)(22 88 50 59)(23 89 51 60)(24 90 52 61)(25 91 53 62)(26 92 54 63)(27 93 28 64)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 81)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)
(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)
(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)(55 104)(56 103)(57 102)(58 101)(59 100)(60 99)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 108)(79 107)(80 106)(81 105)

G:=sub<Sym(108)| (1,94,29,65)(2,95,30,66)(3,96,31,67)(4,97,32,68)(5,98,33,69)(6,99,34,70)(7,100,35,71)(8,101,36,72)(9,102,37,73)(10,103,38,74)(11,104,39,75)(12,105,40,76)(13,106,41,77)(14,107,42,78)(15,108,43,79)(16,82,44,80)(17,83,45,81)(18,84,46,55)(19,85,47,56)(20,86,48,57)(21,87,49,58)(22,88,50,59)(23,89,51,60)(24,90,52,61)(25,91,53,62)(26,92,54,63)(27,93,28,64), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92), (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), (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)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,108)(79,107)(80,106)(81,105)>;

G:=Group( (1,94,29,65)(2,95,30,66)(3,96,31,67)(4,97,32,68)(5,98,33,69)(6,99,34,70)(7,100,35,71)(8,101,36,72)(9,102,37,73)(10,103,38,74)(11,104,39,75)(12,105,40,76)(13,106,41,77)(14,107,42,78)(15,108,43,79)(16,82,44,80)(17,83,45,81)(18,84,46,55)(19,85,47,56)(20,86,48,57)(21,87,49,58)(22,88,50,59)(23,89,51,60)(24,90,52,61)(25,91,53,62)(26,92,54,63)(27,93,28,64), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92), (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), (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)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,108)(79,107)(80,106)(81,105) );

G=PermutationGroup([[(1,94,29,65),(2,95,30,66),(3,96,31,67),(4,97,32,68),(5,98,33,69),(6,99,34,70),(7,100,35,71),(8,101,36,72),(9,102,37,73),(10,103,38,74),(11,104,39,75),(12,105,40,76),(13,106,41,77),(14,107,42,78),(15,108,43,79),(16,82,44,80),(17,83,45,81),(18,84,46,55),(19,85,47,56),(20,86,48,57),(21,87,49,58),(22,88,50,59),(23,89,51,60),(24,90,52,61),(25,91,53,62),(26,92,54,63),(27,93,28,64)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,81),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,93),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,101),(37,102),(38,103),(39,104),(40,105),(41,106),(42,107),(43,108),(44,82),(45,83),(46,84),(47,85),(48,86),(49,87),(50,88),(51,89),(52,90),(53,91),(54,92)], [(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)], [(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),(55,104),(56,103),(57,102),(58,101),(59,100),(60,99),(61,98),(62,97),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,90),(70,89),(71,88),(72,87),(73,86),(74,85),(75,84),(76,83),(77,82),(78,108),(79,107),(80,106),(81,105)]])

75 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 9A 9B 9C 12 18A 18B 18C 18D ··· 18I 27A ··· 27I 36A 36B 36C 54A ··· 54I 54J ··· 54AA 108A ··· 108I order 1 2 2 2 2 2 2 2 3 4 4 6 6 6 9 9 9 12 18 18 18 18 ··· 18 27 ··· 27 36 36 36 54 ··· 54 54 ··· 54 108 ··· 108 size 1 1 2 2 27 27 54 54 2 2 54 2 4 4 2 2 2 4 2 2 2 4 ··· 4 2 ··· 2 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 D9 D18 D18 D27 D54 D54 S3×D4 D4×D9 D4×D27 kernel D4×D27 C4×D27 D108 C27⋊D4 D4×C27 C22×D27 D4×C9 D27 C36 C2×C18 C3×D4 C12 C2×C6 D4 C4 C22 C9 C3 C1 # reps 1 1 1 2 1 2 1 2 1 2 3 3 6 9 9 18 1 3 9

Matrix representation of D4×D27 in GL4(𝔽109) generated by

 1 0 0 0 0 1 0 0 0 0 108 3 0 0 72 1
,
 1 0 0 0 0 1 0 0 0 0 108 3 0 0 0 1
,
 87 51 0 0 58 29 0 0 0 0 1 0 0 0 0 1
,
 82 59 0 0 32 27 0 0 0 0 108 0 0 0 0 108
G:=sub<GL(4,GF(109))| [1,0,0,0,0,1,0,0,0,0,108,72,0,0,3,1],[1,0,0,0,0,1,0,0,0,0,108,0,0,0,3,1],[87,58,0,0,51,29,0,0,0,0,1,0,0,0,0,1],[82,32,0,0,59,27,0,0,0,0,108,0,0,0,0,108] >;

D4×D27 in GAP, Magma, Sage, TeX

D_4\times D_{27}
% in TeX

G:=Group("D4xD27");
// GroupNames label

G:=SmallGroup(432,47);
// by ID

G=gap.SmallGroup(432,47);
# by ID

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

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

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