direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4.D4, M4(2)⋊14D6, (C4×S3).1D4, C4.146(S3×D4), C12.89(C2×D4), (C2×D4).121D6, C12.D4⋊3C2, (S3×M4(2))⋊5C2, (S3×C23).2C4, (C2×C12).1C23, C23.11(C4×S3), C12.46D4⋊9C2, (C6×D4).11C22, C4.Dic3⋊1C22, D6.16(C22⋊C4), (C2×D12).38C22, (C3×M4(2))⋊16C22, Dic3.4(C22⋊C4), (C2×S3×D4).2C2, C3⋊1(C2×C4.D4), (C2×C3⋊D4).1C4, (S3×C2×C4).1C22, C22.14(S3×C2×C4), (C3×C4.D4)⋊9C2, C2.13(S3×C22⋊C4), C6.12(C2×C22⋊C4), (C2×C6).8(C22×C4), (C22×C6).6(C2×C4), (C2×C4).1(C22×S3), (C22×S3).2(C2×C4), (C2×Dic3).19(C2×C4), SmallGroup(192,303)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C4.D4
G = < a,b,c,d,e | a3=b2=c4=1, d4=c2, e2=c, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=c-1d3 >
Subgroups: 656 in 186 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C24, C3⋊C8, C24, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C4.D4, C4.D4, C2×M4(2), C22×D4, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C2×C4.D4, C12.46D4, C12.D4, C3×C4.D4, S3×M4(2), C2×S3×D4, S3×C4.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C4.D4, C2×C22⋊C4, S3×C2×C4, S3×D4, C2×C4.D4, S3×C22⋊C4, S3×C4.D4
►On 24 points - transitive group 24T339
(1 16 22)(2 9 23)(3 10 24)(4 11 17)(5 12 18)(6 13 19)(7 14 20)(8 15 21)
(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 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 6 3 4 5 2 7 8)(9 14 15 16 13 10 11 12)(17 18 23 20 21 22 19 24)
G:=sub<Sym(24)| (1,16,22)(2,9,23)(3,10,24)(4,11,17)(5,12,18)(6,13,19)(7,14,20)(8,15,21), (9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,6,3,4,5,2,7,8)(9,14,15,16,13,10,11,12)(17,18,23,20,21,22,19,24)>;
G:=Group( (1,16,22)(2,9,23)(3,10,24)(4,11,17)(5,12,18)(6,13,19)(7,14,20)(8,15,21), (9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,6,3,4,5,2,7,8)(9,14,15,16,13,10,11,12)(17,18,23,20,21,22,19,24) );
G=PermutationGroup([[(1,16,22),(2,9,23),(3,10,24),(4,11,17),(5,12,18),(6,13,19),(7,14,20),(8,15,21)], [(9,23),(10,24),(11,17),(12,18),(13,19),(14,20),(15,21),(16,22)], [(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16),(17,23,21,19),(18,20,22,24)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,6,3,4,5,2,7,8),(9,14,15,16,13,10,11,12),(17,18,23,20,21,22,19,24)]])
G:=TransitiveGroup(24,339);
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 12 | 12 | 2 | 2 | 2 | 6 | 6 | 2 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D6 | C4×S3 | C4.D4 | S3×D4 | S3×C4.D4 |
| kernel | S3×C4.D4 | C12.46D4 | C12.D4 | C3×C4.D4 | S3×M4(2) | C2×S3×D4 | C2×C3⋊D4 | S3×C23 | C4.D4 | C4×S3 | M4(2) | C2×D4 | C23 | S3 | C4 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 4 | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 1 |
Matrix representation of S3×C4.D4 ►in GL6(𝔽73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 3 | 0 | 0 |
| 0 | 0 | 48 | 1 | 0 | 0 |
| 0 | 0 | 59 | 0 | 0 | 1 |
| 0 | 0 | 59 | 42 | 72 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 42 | 71 | 0 |
| 0 | 0 | 55 | 27 | 48 | 25 |
| 0 | 0 | 60 | 35 | 13 | 14 |
| 0 | 0 | 59 | 4 | 59 | 60 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 42 | 71 | 0 |
| 0 | 0 | 0 | 1 | 48 | 48 |
| 0 | 0 | 0 | 0 | 13 | 59 |
| 0 | 0 | 1 | 0 | 59 | 13 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,48,59,59,0,0,3,1,0,42,0,0,0,0,0,72,0,0,0,0,1,0],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,46,55,60,59,0,0,42,27,35,4,0,0,71,48,13,59,0,0,0,25,14,60],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,1,0,0,42,1,0,0,0,0,71,48,13,59,0,0,0,48,59,13] >;
S3×C4.D4 in GAP, Magma, Sage, TeX
S_3\times C_4.D_4
% in TeX
G:=Group("S3xC4.D4"); // GroupNames label
G:=SmallGroup(192,303);
// by ID
G=gap.SmallGroup(192,303);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,58,570,136,438,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=1,d^4=c^2,e^2=c,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^-1=c^-1,c*e=e*c,e*d*e^-1=c^-1*d^3>;
// generators/relations