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## G = C3×S3×Dic3order 216 = 23·33

### Direct product of C3, S3 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×S3×Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×S3×Dic3
 Lower central C32 — C3×S3×Dic3
 Upper central C1 — C6

Generators and relations for C3×S3×Dic3
G = < a,b,c,d,e | a3=b3=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 228 in 90 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2 [×2], C3 [×3], C3 [×4], C4 [×2], C22, S3 [×2], C6 [×3], C6 [×10], C2×C4, C32 [×3], C32 [×4], Dic3, Dic3 [×3], C12 [×4], D6, C2×C6 [×3], C3×S3 [×4], C3×S3 [×2], C3×C6 [×3], C3×C6 [×6], C4×S3, C2×Dic3, C2×C12, C33, C3×Dic3 [×2], C3×Dic3 [×4], C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6, C62, S3×C32 [×2], C32×C6, S3×Dic3, S3×C12, C6×Dic3, C32×Dic3, C3×C3⋊Dic3, S3×C3×C6, C3×S3×Dic3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3 [×2], C6 [×3], C2×C4, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C3×S3 [×2], C4×S3, C2×Dic3, C2×C12, C3×Dic3 [×2], S32, S3×C6 [×2], S3×Dic3, S3×C12, C6×Dic3, C3×S32, C3×S3×Dic3

Permutation representations of C3×S3×Dic3
On 24 points - transitive group 24T545
Generators in S24
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 8 4 11)(2 7 5 10)(3 12 6 9)(13 20 16 23)(14 19 17 22)(15 24 18 21)

G:=sub<Sym(24)| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,8,4,11)(2,7,5,10)(3,12,6,9)(13,20,16,23)(14,19,17,22)(15,24,18,21)>;

G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,8,4,11)(2,7,5,10)(3,12,6,9)(13,20,16,23)(14,19,17,22)(15,24,18,21) );

G=PermutationGroup([(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,16),(2,17),(3,18),(4,13),(5,14),(6,15),(7,22),(8,23),(9,24),(10,19),(11,20),(12,21)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,8,4,11),(2,7,5,10),(3,12,6,9),(13,20,16,23),(14,19,17,22),(15,24,18,21)])

G:=TransitiveGroup(24,545);

C3×S3×Dic3 is a maximal subgroup of   (S3×C6).D6  D6.S32  D6.4S32  D6.3S32  S32×C12

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 6A 6B 6C ··· 6H 6I 6J 6K 6L 6M 6N 6O 6P ··· 6U 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 6 6 6 ··· 6 6 6 6 6 6 6 6 6 ··· 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 3 3 1 1 2 ··· 2 4 4 4 3 3 9 9 1 1 2 ··· 2 3 3 3 3 4 4 4 6 ··· 6 3 3 3 3 6 ··· 6 9 9 9 9

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 S3 Dic3 D6 C3×S3 C3×S3 C4×S3 C3×Dic3 S3×C6 S3×C12 S32 S3×Dic3 C3×S32 C3×S3×Dic3 kernel C3×S3×Dic3 C32×Dic3 C3×C3⋊Dic3 S3×C3×C6 S3×Dic3 S3×C32 C3×Dic3 C3⋊Dic3 S3×C6 C3×S3 C3×Dic3 S3×C6 C3×S3 C3×C6 Dic3 D6 C32 S3 C6 C3 C6 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 2 4 4 4 1 1 2 2

Matrix representation of C3×S3×Dic3 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 6 0 3 1 5 3 6 0 1 6 4 5 3 3 2 6
,
 4 2 0 6 1 3 3 2 5 6 0 3 3 0 6 0
,
 5 1 2 1 3 4 6 4 6 6 5 2 6 1 4 2
,
 4 0 5 0 4 6 4 6 5 0 3 0 4 2 0 1
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,5,1,3,0,3,6,3,3,6,4,2,1,0,5,6],[4,1,5,3,2,3,6,0,0,3,0,6,6,2,3,0],[5,3,6,6,1,4,6,1,2,6,5,4,1,4,2,2],[4,4,5,4,0,6,0,2,5,4,3,0,0,6,0,1] >;

C3×S3×Dic3 in GAP, Magma, Sage, TeX

C_3\times S_3\times {\rm Dic}_3
% in TeX

G:=Group("C3xS3xDic3");
// GroupNames label

G:=SmallGroup(216,119);
// by ID

G=gap.SmallGroup(216,119);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,79,730,5189]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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