direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4≀C2, C42⋊33D6, M4(2)⋊15D6, D4⋊7(C4×S3), (S3×D4)⋊3C4, (S3×Q8)⋊3C4, Q8⋊8(C4×S3), D12⋊5(C2×C4), D4⋊2S3⋊3C4, (S3×C42)⋊1C2, (C4×C12)⋊9C22, Q8⋊3S3⋊3C4, C4○D4.34D6, Dic6⋊5(C2×C4), (C4×S3).32D4, C4.200(S3×D4), D12⋊C4⋊9C2, C42⋊4S3⋊3C2, C12.359(C2×D4), (S3×M4(2))⋊8C2, C22.27(S3×D4), Q8⋊3Dic3⋊1C2, C12.17(C22×C4), C4○D12.9C22, (C22×S3).80D4, C4.Dic3⋊2C22, (C2×C12).260C23, D6.20(C22⋊C4), (C2×Dic3).159D4, (C4×Dic3)⋊61C22, (C3×M4(2))⋊17C22, Dic3.10(C22⋊C4), C3⋊1(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
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, D4⋊2S3, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, C2×C4≀C2, C42⋊4S3, D12⋊C4, Q8⋊3Dic3, 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
►On 24 points - transitive group 24T361
(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