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G = C2xD12  order 48 = 24·3

Direct product of C2 and D12

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

Derived series
C1 — C6 — C2xD12
Chief series
C1 — C3 — C6 — D6 — C22xS3 — C2xD12
Lower central
C3 — C6 — C2xD12
Upper central
C1 — C22 — C2xC4

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 12A2B2C2D2E2F2G34A4B6A6B6C12A12B12C12D
 size 111166662222222222
ρ1111111111111111111    trivial
ρ21-1-1111-1-111-1-1-11-111-1    linear of order 2
ρ31-1-11-1-11111-1-1-11-111-1    linear of order 2
ρ41-1-111-1-111-11-1-111-1-11    linear of order 2
ρ51-1-11-111-11-11-1-111-1-11    linear of order 2
ρ61111-1-1-1-11111111111    linear of order 2
ρ711111-11-11-1-1111-1-1-1-1    linear of order 2
ρ81111-11-111-1-1111-1-1-1-1    linear of order 2
ρ922220000-1-2-2-1-1-11111    orthogonal lifted from D6
ρ1022220000-122-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1122-2-200002002-2-20000    orthogonal lifted from D4
ρ122-2-220000-12-211-11-1-11    orthogonal lifted from D6
ρ132-2-220000-1-2211-1-111-1    orthogonal lifted from D6
ρ142-22-20000200-22-20000    orthogonal lifted from D4
ρ1522-2-20000-100-111√3-√3√3-√3    orthogonal lifted from D12
ρ1622-2-20000-100-111-√3√3-√3√3    orthogonal lifted from D12
ρ172-22-20000-1001-11√3√3-√3-√3    orthogonal lifted from D12
ρ182-22-20000-1001-11-√3-√3√3√3    orthogonal lifted from D12

Permutation representations of C2xD12
►On 24 points - transitive group 24T29
Generators in S24
(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

1200
010
001
,
100
0310
036
,
1200
0310
0710
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
 

Export

Character table of C2xD12 in TeX

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