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## G = S3×C4≀C2order 192 = 26·3

### Direct product of S3 and C4≀C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — S3×C4≀C2
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — S3×C4○D4 — S3×C4≀C2
 Lower central C3 — C6 — C12 — S3×C4≀C2
 Upper central C1 — C4 — C2×C4 — C4≀C2

Generators and relations for S3×C4≀C2
G = < a,b,c,d,e | a3=b2=c4=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c-1d >

Subgroups: 480 in 170 conjugacy classes, 53 normal (51 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C4≀C2, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, S3×C8, C8⋊S3, C4.Dic3, C4×Dic3, C4×Dic3, C4×C12, C3×M4(2), S3×C2×C4, S3×C2×C4, C4○D12, C4○D12, S3×D4, S3×D4, D42S3, D42S3, S3×Q8, Q83S3, C3×C4○D4, C2×C4≀C2, C424S3, D12⋊C4, Q83Dic3, C3×C4≀C2, S3×C42, S3×M4(2), S3×C4○D4, S3×C4≀C2
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C4≀C2, C2×C22⋊C4, S3×C2×C4, S3×D4, C2×C4≀C2, S3×C22⋊C4, S3×C4≀C2

Permutation representations of S3×C4≀C2
On 24 points - transitive group 24T361
Generators in S24
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 20)(14 17)(15 18)(16 19)(21 23)(22 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 21)(2 24)(3 23)(4 22)(5 19)(6 18)(7 17)(8 20)(9 15)(10 14)(11 13)(12 16)
(1 4 3 2)(13 16 15 14)(17 20 19 18)

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

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

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

G:=TransitiveGroup(24,361);

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C ··· 4G 4H 4I 4J 4K ··· 4O 4P 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C ··· 12G 12H 24A 24B order 1 2 2 2 2 2 2 2 3 4 4 4 ··· 4 4 4 4 4 ··· 4 4 6 6 6 8 8 8 8 12 12 12 ··· 12 12 24 24 size 1 1 2 3 3 4 6 12 2 1 1 2 ··· 2 3 3 4 6 ··· 6 12 2 4 8 4 4 12 12 2 2 4 ··· 4 8 8 8

42 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 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D4 D4 D4 D6 D6 D6 C4×S3 C4×S3 C4≀C2 S3×D4 S3×D4 S3×C4≀C2 kernel S3×C4≀C2 C42⋊4S3 D12⋊C4 Q8⋊3Dic3 C3×C4≀C2 S3×C42 S3×M4(2) S3×C4○D4 S3×D4 D4⋊2S3 S3×Q8 Q8⋊3S3 C4≀C2 C4×S3 C2×Dic3 C22×S3 C42 M4(2) C4○D4 D4 Q8 S3 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 1 2 1 1 1 1 1 2 2 8 1 1 4

Matrix representation of S3×C4≀C2 in GL4(𝔽5) generated by

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

S3×C4≀C2 in GAP, Magma, Sage, TeX

S_3\times C_4\wr C_2
% in TeX

G:=Group("S3xC4wrC2");
// GroupNames label

G:=SmallGroup(192,379);
// by ID

G=gap.SmallGroup(192,379);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,58,136,851,438,102,6278]);
// Polycyclic

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

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