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G = D4xC3xC9order 216 = 23·33

Direct product of C3xC9 and D4

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

Aliases: D4xC3xC9, C36:7C6, C12:3C18, C6.8C62, C62.9C6, C4:(C3xC18), (C2xC6):3C18, (C3xC36):7C2, (C2xC18):9C6, (C6xC18):2C2, C2.1(C6xC18), C6.8(C2xC18), C12.3(C3xC6), C18.16(C2xC6), (C3xC12).17C6, C22:3(C3xC18), C32.4(C3xD4), C3.1(D4xC32), (C3xD4).1C32, (D4xC32).2C3, (C3xC18).23C22, (C2xC6).6(C3xC6), (C3xC6).34(C2xC6), (C3xC18)o(D4xC32), SmallGroup(216,76)

Series: Derived Chief Lower central Upper central

C1C2 — D4xC3xC9
C1C3C6C3xC6C3xC18C6xC18 — D4xC3xC9
C1C2 — D4xC3xC9
C1C3xC18 — D4xC3xC9

Generators and relations for D4xC3xC9
 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, C2xC6, C18, C18, C3xC6, C3xC6, C3xD4, C3xD4, C3xC9, C36, C2xC18, C3xC12, C62, C3xC18, C3xC18, D4xC9, D4xC32, C3xC36, C6xC18, D4xC3xC9
Quotients: C1, C2, C3, C22, C6, D4, C9, C32, C2xC6, C18, C3xC6, C3xD4, C3xC9, C2xC18, C62, C3xC18, D4xC9, D4xC32, C6xC18, D4xC3xC9

Smallest permutation representation of D4xC3xC9
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)]])

D4xC3xC9 is a maximal subgroup of   C36.17D6  C36.18D6  C36.27D6

135 conjugacy classes

class 1 2A2B2C3A···3H 4 6A···6H6I···6X9A···9R12A···12H18A···18R18S···18BB36A···36R
order12223···346···66···69···912···1218···1818···1836···36
size11221···121···12···21···12···21···12···22···2

135 irreducible representations

dim1111111111112222
type++++
imageC1C2C2C3C3C6C6C6C6C9C18C18D4C3xD4C3xD4D4xC9
kernelD4xC3xC9C3xC36C6xC18D4xC9D4xC32C36C2xC18C3xC12C62C3xD4C12C2xC6C3xC9C9C32C3
# reps112626122418183616218

Matrix representation of D4xC3xC9 in GL3(F37) generated by

2600
0260
0026
,
100
0120
0012
,
3600
0036
010
,
3600
010
0036
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] >;

D4xC3xC9 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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