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G = C2×D12  order 48 = 24·3

Direct product of C2 and D12

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D12, C4⋊2D6, C6⋊1D4, C12⋊2C22, D6⋊1C22, C6.3C23, C22.10D6, C3⋊1(C2×D4), (C2×C4)⋊2S3, (C2×C12)⋊3C2, (C22×S3)⋊1C2, C2.4(C22×S3), (C2×C6).10C22, SmallGroup(48,36)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C2×D12
Chief series
C1 — C3 — C6 — D6 — C22×S3 — C2×D12
Lower central
C3 — C6 — C2×D12
Upper central
C1 — C22 — C2×C4

Generators and relations for C2×D12
 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, C2×C4, D4, C23, C12, D6, D6, C2×C6, C2×D4, D12, C2×C12, C22×S3, C2×D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C2×D12

Character table of C2×D12

 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 C2×D12
►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);
 

C2×D12 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  C2×S3×D4  D4○D12  Q8⋊D12
C2×D12 is a maximal quotient of
 C12⋊2Q8  C4⋊D12  C42⋊7S3  D6⋊D4  C23.21D6  C12⋊D4  C4.D12  C4○D24  C8⋊D6  C8.D6  C12⋊7D4

Matrix representation of C2×D12 ►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] >;
 

C2×D12 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 C2×D12 in TeX

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