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## G = C9×C3⋊D4order 216 = 23·33

### Direct product of C9 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C9×C3⋊D4
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — S3×C18 — C9×C3⋊D4
 Lower central C3 — C6 — C9×C3⋊D4
 Upper central C1 — C18 — C2×C18

Generators and relations for C9×C3⋊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 [×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], C18, C18 [×5], C3×S3, C3×C6, C3×C6, C3⋊D4, C3×D4, C3×C9, C36, C2×C18, C2×C18 [×2], C3×Dic3, S3×C6, C62, S3×C9, C3×C18, C3×C18, D4×C9, C3×C3⋊D4, C9×Dic3, S3×C18, C6×C18, C9×C3⋊D4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, C9, D6, C2×C6, C18 [×3], C3×S3, C3⋊D4, C3×D4, C2×C18, S3×C6, S3×C9, D4×C9, C3×C3⋊D4, S3×C18, C9×C3⋊D4

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

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

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

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

C9×C3⋊D4 is a maximal subgroup of   Dic3.D18  D18.4D6  D18⋊D6  S3×D4×C9

81 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4 6A 6B 6C ··· 6M 6N 6O 9A ··· 9F 9G ··· 9L 12A 12B 18A ··· 18F 18G ··· 18AD 18AE ··· 18AJ 36A ··· 36F order 1 2 2 2 3 3 3 3 3 4 6 6 6 ··· 6 6 6 9 ··· 9 9 ··· 9 12 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 6 1 1 2 2 2 6 1 1 2 ··· 2 6 6 1 ··· 1 2 ··· 2 6 6 1 ··· 1 2 ··· 2 6 ··· 6 6 ··· 6

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 S3×C9 D4×C9 C3×C3⋊D4 S3×C18 C9×C3⋊D4 kernel C9×C3⋊D4 C9×Dic3 S3×C18 C6×C18 C3×C3⋊D4 C3×Dic3 S3×C6 C62 C3⋊D4 Dic3 D6 C2×C6 C2×C18 C3×C9 C18 C2×C6 C9 C32 C6 C22 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 6 6 6 6 1 1 1 2 2 2 2 6 6 4 6 12

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

 17 0 0 17
,
 9 2 2 9
,
 0 1 18 0
,
 18 0 0 1
G:=sub<GL(2,GF(19))| [17,0,0,17],[9,2,2,9],[0,18,1,0],[18,0,0,1] >;

C9×C3⋊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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