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## G = D4×C3×C9order 216 = 23·33

### Direct product of C3×C9 and D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C3×C9
 Chief series C1 — C3 — C6 — C3×C6 — C3×C18 — C6×C18 — D4×C3×C9
 Lower central C1 — C2 — D4×C3×C9
 Upper central C1 — C3×C18 — D4×C3×C9

Generators and relations for D4×C3×C9
G = < a,b,c,d | a3=b9=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 100 in 80 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, D4, C9, C32, C12, C12, C2×C6, C18, C18, C3×C6, C3×C6, C3×D4, C3×D4, C3×C9, C36, C2×C18, C3×C12, C62, C3×C18, C3×C18, D4×C9, D4×C32, C3×C36, C6×C18, D4×C3×C9
Quotients: C1, C2, C3, C22, C6, D4, C9, C32, C2×C6, C18, C3×C6, C3×D4, C3×C9, C2×C18, C62, C3×C18, D4×C9, D4×C32, C6×C18, D4×C3×C9

Smallest permutation representation of D4×C3×C9
On 108 points
Generators in S108
(1 59 16)(2 60 17)(3 61 18)(4 62 10)(5 63 11)(6 55 12)(7 56 13)(8 57 14)(9 58 15)(19 72 103)(20 64 104)(21 65 105)(22 66 106)(23 67 107)(24 68 108)(25 69 100)(26 70 101)(27 71 102)(28 51 80)(29 52 81)(30 53 73)(31 54 74)(32 46 75)(33 47 76)(34 48 77)(35 49 78)(36 50 79)(37 97 87)(38 98 88)(39 99 89)(40 91 90)(41 92 82)(42 93 83)(43 94 84)(44 95 85)(45 96 86)
(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)
(1 101 36 97)(2 102 28 98)(3 103 29 99)(4 104 30 91)(5 105 31 92)(6 106 32 93)(7 107 33 94)(8 108 34 95)(9 100 35 96)(10 64 73 40)(11 65 74 41)(12 66 75 42)(13 67 76 43)(14 68 77 44)(15 69 78 45)(16 70 79 37)(17 71 80 38)(18 72 81 39)(19 52 89 61)(20 53 90 62)(21 54 82 63)(22 46 83 55)(23 47 84 56)(24 48 85 57)(25 49 86 58)(26 50 87 59)(27 51 88 60)
(19 89)(20 90)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(37 70)(38 71)(39 72)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(91 104)(92 105)(93 106)(94 107)(95 108)(96 100)(97 101)(98 102)(99 103)

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

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

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

D4×C3×C9 is a maximal subgroup of   C36.17D6  C36.18D6  C36.27D6

135 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4 6A ··· 6H 6I ··· 6X 9A ··· 9R 12A ··· 12H 18A ··· 18R 18S ··· 18BB 36A ··· 36R order 1 2 2 2 3 ··· 3 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 2 1 ··· 1 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 C9 C18 C18 D4 C3×D4 C3×D4 D4×C9 kernel D4×C3×C9 C3×C36 C6×C18 D4×C9 D4×C32 C36 C2×C18 C3×C12 C62 C3×D4 C12 C2×C6 C3×C9 C9 C32 C3 # reps 1 1 2 6 2 6 12 2 4 18 18 36 1 6 2 18

Matrix representation of D4×C3×C9 in GL3(𝔽37) generated by

 26 0 0 0 26 0 0 0 26
,
 1 0 0 0 12 0 0 0 12
,
 36 0 0 0 0 36 0 1 0
,
 36 0 0 0 1 0 0 0 36
G:=sub<GL(3,GF(37))| [26,0,0,0,26,0,0,0,26],[1,0,0,0,12,0,0,0,12],[36,0,0,0,0,1,0,36,0],[36,0,0,0,1,0,0,0,36] >;

D4×C3×C9 in GAP, Magma, Sage, TeX

D_4\times C_3\times C_9
% in TeX

G:=Group("D4xC3xC9");
// GroupNames label

G:=SmallGroup(216,76);
// by ID

G=gap.SmallGroup(216,76);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,457,338]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=c^4=d^2=1,a*b=b*a,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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