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G = D6⋊C4  order 48 = 24·3

The semidirect product of D6 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊C4, C6.6D4, C2.2D12, C22.6D6, (C2×C4)⋊1S3, (C2×C12)⋊1C2, C2.5(C4×S3), C6.4(C2×C4), C3⋊1(C22⋊C4), (C22×S3).C2, (C2×Dic3)⋊1C2, C2.2(C3⋊D4), (C2×C6).6C22, SmallGroup(48,14)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for D6⋊C4
 G = < a,b,c | a6=b2=c4=1, bab=a-1, ac=ca, cbc-1=a3b >

C1
C2
C2
C2
6C2
6C2
C3
C22
2C4
3C22
3C22
6C4
6C22
6C22
C6
C6
C6
2S3
2S3
C2×C4
3C2×C4
3C23
C2×C6
D6
D6
2D6
2D6
2Dic3
2C12
3C22⋊C4
C2×Dic3
C2×C12
C22×S3
D6⋊C4

Character table of D6⋊C4

 class 12A2B2C2D2E34A4B4C4D6A6B6C12A12B12C12D
 size 111166222662222222
ρ1111111111111111111    trivial
ρ21111-1-11-1-111111-1-1-1-1    linear of order 2
ρ31111-1-1111-1-11111111    linear of order 2
ρ41111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ51-1-111-11i-i-ii1-1-1-i-iii    linear of order 4
ρ61-1-11-111i-ii-i1-1-1-i-iii    linear of order 4
ρ71-1-11-111-ii-ii1-1-1ii-i-i    linear of order 4
ρ81-1-111-11-iii-i1-1-1ii-i-i    linear of order 4
ρ9222200-12200-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ10222200-1-2-200-1-1-11111    orthogonal lifted from D6
ρ1122-2-20020000-22-20000    orthogonal lifted from D4
ρ122-22-20020000-2-220000    orthogonal lifted from D4
ρ1322-2-200-100001-11-√3√3√3-√3    orthogonal lifted from D12
ρ1422-2-200-100001-11√3-√3-√3√3    orthogonal lifted from D12
ρ152-2-2200-1-2i2i00-111-i-iii    complex lifted from C4×S3
ρ162-2-2200-12i-2i00-111ii-i-i    complex lifted from C4×S3
ρ172-22-200-1000011-1-√-3√-3-√-3√-3    complex lifted from C3⋊D4
ρ182-22-200-1000011-1√-3-√-3√-3-√-3    complex lifted from C3⋊D4

Permutation representations of D6⋊C4
►On 24 points - transitive group 24T33
Generators in S24
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 21)(14 20)(15 19)(16 24)(17 23)(18 22)

(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)
 
G:=sub<Sym(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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)>;
 
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18) );
 
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,21),(14,20),(15,19),(16,24),(17,23),(18,22)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18)]])
 
G:=TransitiveGroup(24,33);
 

D6⋊C4 is a maximal subgroup of
 C42⋊2S3  C4×D12  C42⋊7S3  C42⋊3S3  S3×C22⋊C4  Dic3⋊4D4  D6⋊D4  C23.9D6  Dic3⋊D4  C23.11D6  C23.21D6  C4⋊C4⋊7S3  Dic3⋊5D4  D6.D4  C12⋊D4  D6⋊Q8  C4.D12  C4⋊C4⋊S3  C4×C3⋊D4  C23.28D6  C12⋊7D4  C23⋊2D6  C23.14D6  D6⋊3Q8  C12.23D4  D18⋊C4  D6⋊Dic3  C6.D12  C6.11D12  GL2(𝔽3)⋊C4  Q8.2D12  C24.5D6  SL2(𝔽3)⋊D4  D6⋊Dic5  D30⋊4C4  D30⋊3C4  D6⋊F5  D6⋊Dic7  D42⋊C4  C2.D84  C32⋊D6⋊C4  D6⋊(C32⋊C4)  C3⋊S3.2D12  (C3×C6).8D12
D6⋊C4 is a maximal quotient of
 C42⋊4S3  C23.6D6  C6.D8  C6.SD16  C2.Dic12  D6⋊C8  C2.D24  C12.46D4  C12.47D4  D12⋊C4  C6.C42  D18⋊C4  D6⋊Dic3  C6.D12  C6.11D12  C24.5D6  D6⋊Dic5  D30⋊4C4  D30⋊3C4  D6⋊F5  D6⋊Dic7  D42⋊C4  C2.D84  D6⋊(C32⋊C4)  C3⋊S3.2D12  (C3×C6).8D12

Matrix representation of D6⋊C4 ►in GL3(𝔽13) generated by

100
011
0120
,
1200
011
0012
,
500
0119
042
G:=sub<GL(3,GF(13))| [1,0,0,0,1,12,0,1,0],[12,0,0,0,1,0,0,1,12],[5,0,0,0,11,4,0,9,2] >;
 

D6⋊C4 in GAP, Magma, Sage, TeX 

D_6\rtimes C_4
 
% in TeX
 
G:=Group("D6:C4");
 
// GroupNames label
 
G:=SmallGroup(48,14);
 
// by ID
 
G=gap.SmallGroup(48,14);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-3,101,26,804]);
 
// Polycyclic
 
G:=Group<a,b,c|a^6=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;
 
// generators/relations
 

Export

Subgroup lattice of D6⋊C4 in TeX
Character table of D6⋊C4 in TeX

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