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G = C3×C9⋊D4  order 216 = 23·33

Direct product of C3 and C9⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C9⋊D4, D18⋊5C6, Dic9⋊4C6, C6.23D18, C62.10S3, (C3×C9)⋊8D4, (C2×C6)⋊3D9, C9⋊5(C3×D4), (C6×C18)⋊4C2, (C6×D9)⋊4C2, C2.5(C6×D9), (C2×C18)⋊10C6, C6.16(S3×C6), (C3×C6).49D6, C22⋊3(C3×D9), C18.13(C2×C6), (C3×Dic9)⋊4C2, (C3×C18).17C22, C32.4(C3⋊D4), C3.1(C3×C3⋊D4), (C2×C6).11(C3×S3), SmallGroup(216,57)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C18 — C3×C9⋊D4
Chief series
C1 — C3 — C9 — C18 — C3×C18 — C6×D9 — C3×C9⋊D4
Lower central
C9 — C18 — C3×C9⋊D4
Upper central
C1 — C6 — C2×C6

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

Subgroups: 180 in 58 conjugacy classes, 24 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, D4, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, D9, C18, C18, C3×S3, C3×C6, C3×C6, C3⋊D4, C3×D4, C3×C9, Dic9, D18, C2×C18, C2×C18, C3×Dic3, S3×C6, C62, C3×D9, C3×C18, C3×C18, C9⋊D4, C3×C3⋊D4, C3×Dic9, C6×D9, C6×C18, C3×C9⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D9, C3×S3, C3⋊D4, C3×D4, D18, S3×C6, C3×D9, C9⋊D4, C3×C3⋊D4, C6×D9, C3×C9⋊D4

Smallest permutation representation of C3×C9⋊D4
►On 36 points
Generators in S36
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)

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

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

C3×C9⋊D4 is a maximal subgroup of   D18.3D6  D18.4D6  D18⋊D6  C3×D4×D9

63 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 6A6B6C···6M6N6O9A···9I12A12B18A···18AA
order1222333334666···6669···9121218···18
size112181122218112···218182···218182···2

63 irreducible representations

dim1111111122222222222222
type+++++++++
imageC1C2C2C2C3C6C6C6S3D4D6D9C3×S3C3×D4C3⋊D4D18S3×C6C3×D9C9⋊D4C3×C3⋊D4C6×D9C3×C9⋊D4
kernelC3×C9⋊D4C3×Dic9C6×D9C6×C18C9⋊D4Dic9D18C2×C18C62C3×C9C3×C6C2×C6C2×C6C9C32C6C6C22C3C3C2C1
# reps11112222111322232664612

Matrix representation of C3×C9⋊D4 ►in GL2(𝔽19) generated by

110
011
,
615
131
,
56
214
,
615
413
G:=sub<GL(2,GF(19))| [11,0,0,11],[6,13,15,1],[5,2,6,14],[6,4,15,13] >;
 

C3×C9⋊D4 in GAP, Magma, Sage, TeX 

C_3\times C_9\rtimes D_4
 
% in TeX
 
G:=Group("C3xC9:D4");
 
// GroupNames label
 
G:=SmallGroup(216,57);
 
// by ID
 
G=gap.SmallGroup(216,57);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,3604,208,5189]);
 
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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