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## G = C3×C9⋊D4order 216 = 23·33

### Direct product of C3 and C9⋊D4

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 [×2], C3 [×2], C3, C4, C22, C22, S3, C6 [×2], C6 [×6], D4, C9, C9, C32, Dic3, C12, D6, C2×C6 [×2], C2×C6 [×2], D9, C18, C18 [×4], 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 [×3], C3, C22, S3, C6 [×3], 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 2A 2B 2C 3A 3B 3C 3D 3E 4 6A 6B 6C ··· 6M 6N 6O 9A ··· 9I 12A 12B 18A ··· 18AA order 1 2 2 2 3 3 3 3 3 4 6 6 6 ··· 6 6 6 9 ··· 9 12 12 18 ··· 18 size 1 1 2 18 1 1 2 2 2 18 1 1 2 ··· 2 18 18 2 ··· 2 18 18 2 ··· 2

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D9 C3×S3 C3×D4 C3⋊D4 D18 S3×C6 C3×D9 C9⋊D4 C3×C3⋊D4 C6×D9 C3×C9⋊D4 kernel C3×C9⋊D4 C3×Dic9 C6×D9 C6×C18 C9⋊D4 Dic9 D18 C2×C18 C62 C3×C9 C3×C6 C2×C6 C2×C6 C9 C32 C6 C6 C22 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 3 2 2 2 3 2 6 6 4 6 12

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

 11 0 0 11
,
 6 15 13 1
,
 5 6 2 14
,
 6 15 4 13
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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