direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3xC2xC4, C12:3C22, C6.2C23, C22.9D6, D6.4C22, Dic3:3C22, C4o(C4xS3), C6:1(C2xC4), (C2xC12):5C2, C3:1(C22xC4), (C2xC4)oDic3, C4o(C2xDic3), (C2xDic3):5C2, C2.1(C22xS3), (C2xC6).9C22, (C22xS3).2C2, (C2xC4)o(C2xDic3), SmallGroup(48,35)
Series: Derived ►Chief ►Lower central ►Upper central
C3 — S3xC2xC4 |
Generators and relations for S3xC2xC4
G = < a,b,c,d | a2=b4=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 92 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C23, Dic3, C12, D6, C2xC6, C22xC4, C4xS3, C2xDic3, C2xC12, C22xS3, S3xC2xC4
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C4xS3, C22xS3, S3xC2xC4
Character table of S3xC2xC4
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ7 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ9 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | i | i | -i | -i | -1 | 1 | -1 | i | -i | i | -i | linear of order 4 |
ρ10 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | -i | i | i | i | -i | -i | -1 | 1 | -1 | -i | i | -i | i | linear of order 4 |
ρ11 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | i | i | -i | -i | -i | i | -i | i | -1 | -1 | 1 | i | -i | -i | i | linear of order 4 |
ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | -i | i | i | -i | i | -i | i | -1 | -1 | 1 | -i | i | i | -i | linear of order 4 |
ρ13 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | i | -i | -i | -i | i | i | -1 | 1 | -1 | i | -i | i | -i | linear of order 4 |
ρ14 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | -i | -i | i | i | -1 | 1 | -1 | -i | i | -i | i | linear of order 4 |
ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | i | -i | -i | i | -i | i | -i | -1 | -1 | 1 | i | -i | -i | i | linear of order 4 |
ρ16 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | -i | i | i | i | -i | i | -i | -1 | -1 | 1 | -i | i | i | -i | linear of order 4 |
ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | i | i | -i | complex lifted from C4xS3 |
ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | i | -i | i | -i | complex lifted from C4xS3 |
ρ23 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -i | i | -i | i | complex lifted from C4xS3 |
ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | -i | -i | i | complex lifted from C4xS3 |
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(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 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 20)(14 17)(15 18)(16 19)(21 23)(22 24)
G:=sub<Sym(24)| (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)>;
G:=Group( (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24) );
G=PermutationGroup([[(1,22),(2,23),(3,24),(4,21),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,20),(14,17),(15,18),(16,19),(21,23),(22,24)]])
G:=TransitiveGroup(24,27);
S3xC2xC4 is a maximal subgroup of
D6:C8 C42:2S3 Dic3:4D4 C23.9D6 Dic3:D4 C4:C4:7S3 Dic3:5D4 D6.D4 C12:D4 D6:Q8 C4.D12 D6:3D4 D6:3Q8
S3xC2xC4 is a maximal quotient of
C42:2S3 C23.16D6 Dic3:4D4 Dic6:C4 C4:C4:7S3 Dic3:5D4 C8oD12 D12.C4
Matrix representation of S3xC2xC4 ►in GL3(F13) generated by
12 | 0 | 0 |
0 | 1 | 0 |
0 | 0 | 1 |
8 | 0 | 0 |
0 | 5 | 0 |
0 | 0 | 5 |
1 | 0 | 0 |
0 | 12 | 12 |
0 | 1 | 0 |
12 | 0 | 0 |
0 | 1 | 0 |
0 | 12 | 12 |
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[8,0,0,0,5,0,0,0,5],[1,0,0,0,12,1,0,12,0],[12,0,0,0,1,12,0,0,12] >;
S3xC2xC4 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_4
% in TeX
G:=Group("S3xC2xC4");
// GroupNames label
G:=SmallGroup(48,35);
// by ID
G=gap.SmallGroup(48,35);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,42,804]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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