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## G = C3×D4⋊S3order 144 = 24·32

### Direct product of C3 and D4⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×D4⋊S3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×D12 — C3×D4⋊S3
 Lower central C3 — C6 — C12 — C3×D4⋊S3
 Upper central C1 — C6 — C12 — C3×D4

Generators and relations for C3×D4⋊S3
G = < a,b,c,d,e | a3=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Permutation representations of C3×D4⋊S3
On 24 points - transitive group 24T246
Generators in S24
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(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 4)(2 3)(6 8)(9 11)(13 14)(15 16)(17 20)(18 19)(22 24)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(1 22)(2 21)(3 24)(4 23)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(24,246);

C3×D4⋊S3 is a maximal subgroup of
Dic63D6  D12⋊D6  D12.D6  D129D6  D12.22D6  D12.8D6  D125D6  C3×S3×D8  He36D8  D36⋊C6  He37D8
C3×D4⋊S3 is a maximal quotient of
He36D8  D36⋊C6

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 6E 6F ··· 6M 6N 6O 8A 8B 12A 12B 12C 12D 12E 24A 24B 24C 24D order 1 2 2 2 3 3 3 3 3 4 6 6 6 6 6 6 ··· 6 6 6 8 8 12 12 12 12 12 24 24 24 24 size 1 1 4 12 1 1 2 2 2 2 1 1 2 2 2 4 ··· 4 12 12 6 6 2 2 4 4 4 6 6 6 6

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D8 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×D8 C3×C3⋊D4 D4⋊S3 C3×D4⋊S3 kernel C3×D4⋊S3 C3×C3⋊C8 C3×D12 D4×C32 D4⋊S3 C3⋊C8 D12 C3×D4 C3×D4 C3×C6 C12 C32 D4 C6 C6 C4 C3 C2 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 2 4 4 1 2

Matrix representation of C3×D4⋊S3 in GL4(𝔽7) generated by

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

C3×D4⋊S3 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes S_3
% in TeX

G:=Group("C3xD4:S3");
// GroupNames label

G:=SmallGroup(144,80);
// by ID

G=gap.SmallGroup(144,80);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,169,867,441,69,3461]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

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