direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xD12, C4:2D6, C6:1D4, C12:2C22, D6:1C22, C6.3C23, C22.10D6, C3:1(C2xD4), (C2xC4):2S3, (C2xC12):3C2, (C22xS3):1C2, C2.4(C22xS3), (C2xC6).10C22, SmallGroup(48,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xD12
G = < a,b,c | a2=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 124 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C12, D6, D6, C2xC6, C2xD4, D12, C2xC12, C22xS3, C2xD12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C22xS3, C2xD12
Character table of C2xD12
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 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 | trivial |
ρ2 | 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 | linear of order 2 |
ρ4 | 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 | linear of order 2 |
ρ6 | 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 | linear of order 2 |
ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ12 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -1 | 1 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -1 | 1 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 21)(14 20)(15 19)(16 18)(22 24)
G:=sub<Sym(24)| (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24)>;
G:=Group( (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,3)(4,12)(5,11)(6,10)(7,9)(13,21)(14,20)(15,19)(16,18)(22,24) );
G=PermutationGroup([[(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,21),(14,20),(15,19),(16,18),(22,24)]])
G:=TransitiveGroup(24,29);
C2xD12 is a maximal subgroup of
C6.D8 C2.D24 C12.46D4 C4:D12 C42:7S3 D6:D4 Dic3:D4 Dic3:5D4 D6.D4 C12:D4 C8:D6 C12:7D4 C12:3D4 C12.23D4 D4:D6 C2xS3xD4 D4oD12 Q8:D12
C2xD12 is a maximal quotient of
C12:2Q8 C4:D12 C42:7S3 D6:D4 C23.21D6 C12:D4 C4.D12 C4oD24 C8:D6 C8.D6 C12:7D4
Matrix representation of C2xD12 ►in GL3(F13) generated by
12 | 0 | 0 |
0 | 1 | 0 |
0 | 0 | 1 |
1 | 0 | 0 |
0 | 3 | 10 |
0 | 3 | 6 |
12 | 0 | 0 |
0 | 3 | 10 |
0 | 7 | 10 |
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[1,0,0,0,3,3,0,10,6],[12,0,0,0,3,7,0,10,10] >;
C2xD12 in GAP, Magma, Sage, TeX
C_2\times D_{12}
% in TeX
G:=Group("C2xD12");
// GroupNames label
G:=SmallGroup(48,36);
// by ID
G=gap.SmallGroup(48,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,182,42,804]);
// Polycyclic
G:=Group<a,b,c|a^2=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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