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G = S3×C4≀C2order 192 = 26·3

Direct product of S3 and C4≀C2

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×C4≀C2, C4233D6, M4(2)⋊15D6, D47(C4×S3), (S3×D4)⋊3C4, (S3×Q8)⋊3C4, Q88(C4×S3), D125(C2×C4), D42S33C4, (S3×C42)⋊1C2, (C4×C12)⋊9C22, Q83S33C4, C4○D4.34D6, Dic65(C2×C4), (C4×S3).32D4, C4.200(S3×D4), D12⋊C49C2, C424S33C2, C12.359(C2×D4), (S3×M4(2))⋊8C2, C22.27(S3×D4), Q83Dic31C2, C12.17(C22×C4), C4○D12.9C22, (C22×S3).80D4, C4.Dic32C22, (C2×C12).260C23, D6.20(C22⋊C4), (C2×Dic3).159D4, (C4×Dic3)⋊61C22, (C3×M4(2))⋊17C22, Dic3.10(C22⋊C4), C31(C2×C4≀C2), (C3×C4≀C2)⋊9C2, C4.17(S3×C2×C4), (C3×D4)⋊5(C2×C4), (C3×Q8)⋊5(C2×C4), (S3×C4○D4).1C2, (C2×C6).24(C2×D4), (C4×S3).16(C2×C4), C6.24(C2×C22⋊C4), C2.25(S3×C22⋊C4), (S3×C2×C4).230C22, (C3×C4○D4).1C22, (C2×C4).367(C22×S3), SmallGroup(192,379)

Series: Derived Chief Lower central Upper central

C1C12 — S3×C4≀C2
C1C3C6C12C2×C12S3×C2×C4S3×C4○D4 — S3×C4≀C2
C3C6C12 — S3×C4≀C2
C1C4C2×C4C4≀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 [×6], C3, C4 [×2], C4 [×8], C22, C22 [×8], S3 [×2], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×16], D4, D4 [×6], Q8, Q8 [×2], C23 [×2], Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×2], D6 [×5], C2×C6, C2×C6, C42, C42 [×2], C2×C8, M4(2), M4(2) [×2], C22×C4 [×3], C2×D4 [×2], C2×Q8, C4○D4, C4○D4 [×5], C3⋊C8, C24, Dic6, Dic6, C4×S3 [×4], C4×S3 [×7], D12, D12, C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C4≀C2, C4≀C2 [×3], 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 [×2], 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 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C4≀C2 [×2], C2×C22⋊C4, S3×C2×C4, S3×D4 [×2], 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 2A2B2C2D2E2F2G 3 4A4B4C···4G4H4I4J4K···4O4P6A6B6C8A8B8C8D12A12B12C···12G12H24A24B
order122222223444···44444···446668888121212···12122424
size1123346122112···23346···612248441212224···4888

42 irreducible representations

dim1111111111112222222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4S3D4D4D4D6D6D6C4×S3C4×S3C4≀C2S3×D4S3×D4S3×C4≀C2
kernelS3×C4≀C2C424S3D12⋊C4Q83Dic3C3×C4≀C2S3×C42S3×M4(2)S3×C4○D4S3×D4D42S3S3×Q8Q83S3C4≀C2C4×S3C2×Dic3C22×S3C42M4(2)C4○D4D4Q8S3C4C22C1
# reps1111111122221211111228114

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

3320
0303
1213
0401
,
2013
1421
1402
2214
,
4002
2420
2413
4001
,
0303
1230
1334
1320
,
3002
2320
2403
4000
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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