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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×Dic6 — 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=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Permutation representations of C3×D4.S3
On 24 points - transitive group 24T245
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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 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), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)>;

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

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

G:=TransitiveGroup(24,245);

C3×D4.S3 is a maximal subgroup of
Dic6.19D6  Dic6⋊D6  Dic6.D6  D12.7D6  Dic6.20D6  D12.8D6  D125D6  C3×S3×SD16  He38SD16  Dic18⋊C6  He39SD16
C3×D4.S3 is a maximal quotient of
He38SD16  Dic18⋊C6

36 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F ··· 6M 8A 8B 12A 12B 12C 12D 12E 12F 12G 24A 24B 24C 24D order 1 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 ··· 6 8 8 12 12 12 12 12 12 12 24 24 24 24 size 1 1 4 1 1 2 2 2 2 12 1 1 2 2 2 4 ··· 4 6 6 2 2 4 4 4 12 12 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 SD16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×SD16 C3×C3⋊D4 D4.S3 C3×D4.S3 kernel C3×D4.S3 C3×C3⋊C8 C3×Dic6 D4×C32 D4.S3 C3⋊C8 Dic6 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

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

C3×D4.S3 in GAP, Magma, Sage, TeX

C_3\times D_4.S_3
% in TeX

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

G:=SmallGroup(144,81);
// by ID

G=gap.SmallGroup(144,81);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,169,867,441,69,3461]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^3=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

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