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G = C9xC3:D4order 216 = 23·33

Direct product of C9 and C3:D4

direct product, metabelian, supersoluble, monomial

Aliases: C9xC3:D4, D6:2C18, Dic3:C18, C18.23D6, C62.11C6, (C3xC9):7D4, C3:2(D4xC9), (C2xC6):4C18, (C2xC18):3S3, (C6xC18):1C2, (S3xC18):2C2, (S3xC6).2C6, C2.5(S3xC18), C6.33(S3xC6), C6.5(C2xC18), C22:3(S3xC9), (C9xDic3):4C2, C32.3(C3xD4), (C3xDic3).3C6, (C3xC18).12C22, (C3xC3:D4).C3, (C2xC6).9(C3xS3), C3.4(C3xC3:D4), (C3xC6).22(C2xC6), SmallGroup(216,58)

Series: Derived Chief Lower central Upper central

C1C6 — C9xC3:D4
C1C3C32C3xC6C3xC18S3xC18 — C9xC3:D4
C3C6 — C9xC3:D4
C1C18C2xC18

Generators and relations for C9xC3:D4
 G = < a,b,c,d | a9=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 110 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, D4, C9, C9, C32, Dic3, C12, D6, C2xC6, C2xC6, C18, C18, C3xS3, C3xC6, C3xC6, C3:D4, C3xD4, C3xC9, C36, C2xC18, C2xC18, C3xDic3, S3xC6, C62, S3xC9, C3xC18, C3xC18, D4xC9, C3xC3:D4, C9xDic3, S3xC18, C6xC18, C9xC3:D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C9, D6, C2xC6, C18, C3xS3, C3:D4, C3xD4, C2xC18, S3xC6, S3xC9, D4xC9, C3xC3:D4, S3xC18, C9xC3:D4

Smallest permutation representation of C9xC3:D4
On 36 points
Generators in S36
(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)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)
(1 14 23 34)(2 15 24 35)(3 16 25 36)(4 17 26 28)(5 18 27 29)(6 10 19 30)(7 11 20 31)(8 12 21 32)(9 13 22 33)
(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)

G:=sub<Sym(36)| (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), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36), (1,14,23,34)(2,15,24,35)(3,16,25,36)(4,17,26,28)(5,18,27,29)(6,10,19,30)(7,11,20,31)(8,12,21,32)(9,13,22,33), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)>;

G:=Group( (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), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36), (1,14,23,34)(2,15,24,35)(3,16,25,36)(4,17,26,28)(5,18,27,29)(6,10,19,30)(7,11,20,31)(8,12,21,32)(9,13,22,33), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27) );

G=PermutationGroup([[(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)], [(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36)], [(1,14,23,34),(2,15,24,35),(3,16,25,36),(4,17,26,28),(5,18,27,29),(6,10,19,30),(7,11,20,31),(8,12,21,32),(9,13,22,33)], [(1,34),(2,35),(3,36),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27)]])

C9xC3:D4 is a maximal subgroup of   Dic3.D18  D18.4D6  D18:D6  S3xD4xC9

81 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 6A6B6C···6M6N6O9A···9F9G···9L12A12B18A···18F18G···18AD18AE···18AJ36A···36F
order1222333334666···6669···99···9121218···1818···1818···1836···36
size1126112226112···2661···12···2661···12···26···66···6

81 irreducible representations

dim111111111111222222222222
type+++++++
imageC1C2C2C2C3C6C6C6C9C18C18C18S3D4D6C3xS3C3:D4C3xD4S3xC6S3xC9D4xC9C3xC3:D4S3xC18C9xC3:D4
kernelC9xC3:D4C9xDic3S3xC18C6xC18C3xC3:D4C3xDic3S3xC6C62C3:D4Dic3D6C2xC6C2xC18C3xC9C18C2xC6C9C32C6C22C3C3C2C1
# reps1111222266661112222664612

Matrix representation of C9xC3:D4 in GL2(F19) generated by

170
017
,
92
29
,
01
180
,
180
01
G:=sub<GL(2,GF(19))| [17,0,0,17],[9,2,2,9],[0,18,1,0],[18,0,0,1] >;

C9xC3:D4 in GAP, Magma, Sage, TeX

C_9\times C_3\rtimes D_4
% in TeX

G:=Group("C9xC3:D4");
// GroupNames label

G:=SmallGroup(216,58);
// by ID

G=gap.SmallGroup(216,58);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,122,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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