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## G = C3×D6⋊S3order 216 = 23·33

### Direct product of C3 and D6⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×D6⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×D6⋊S3
 Lower central C32 — C3×C6 — C3×D6⋊S3
 Upper central C1 — C6

Generators and relations for C3×D6⋊S3
G = < a,b,c,d,e | a3=b6=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 276 in 94 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, D4, C32, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3×C6, C3×C6, C3×C6, C3⋊D4, C3×D4, C33, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, S3×C32, C32×C6, D6⋊S3, C3×C3⋊D4, C3×C3⋊Dic3, S3×C3×C6, C3×D6⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, C3×S3, C3⋊D4, C3×D4, S32, S3×C6, D6⋊S3, C3×C3⋊D4, C3×S32, C3×D6⋊S3

Permutation representations of C3×D6⋊S3
On 24 points - transitive group 24T546
Generators in S24
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)
(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 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)

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

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

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

G:=TransitiveGroup(24,546);

C3×D6⋊S3 is a maximal subgroup of
C33⋊D8  C337SD16  D64S32  (S3×C6)⋊D6  (S3×C6).D6  D6.4S32  D6⋊S3⋊S3  C3×S3×C3⋊D4

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4 6A 6B 6C ··· 6H 6I 6J 6K 6L ··· 6AA 12A 12B order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 6 6 6 ··· 6 6 6 6 6 ··· 6 12 12 size 1 1 6 6 1 1 2 ··· 2 4 4 4 18 1 1 2 ··· 2 4 4 4 6 ··· 6 18 18

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - image C1 C2 C2 C3 C6 C6 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×C3⋊D4 S32 D6⋊S3 C3×S32 C3×D6⋊S3 kernel C3×D6⋊S3 C3×C3⋊Dic3 S3×C3×C6 D6⋊S3 C3⋊Dic3 S3×C6 S3×C6 C33 C3×C6 D6 C32 C32 C6 C3 C6 C3 C2 C1 # reps 1 1 2 2 2 4 2 1 2 4 4 2 4 8 1 1 2 2

Matrix representation of C3×D6⋊S3 in GL4(𝔽7) generated by

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

C3×D6⋊S3 in GAP, Magma, Sage, TeX

C_3\times D_6\rtimes S_3
% in TeX

G:=Group("C3xD6:S3");
// GroupNames label

G:=SmallGroup(216,121);
// by ID

G=gap.SmallGroup(216,121);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,730,5189]);
// Polycyclic

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

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