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G = C4○D12order 48 = 24·3

Central product of C4 and D12

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4D12, C4Dic6, D125C2, C4.16D6, Dic65C2, C6.4C23, C22.2D6, D6.1C22, C12.16C22, Dic3.2C22, (C2×C4)⋊3S3, (C4×S3)⋊4C2, (C2×C12)⋊4C2, C4(C3⋊D4), C31(C4○D4), C3⋊D43C2, C2.5(C22×S3), (C2×C6).11C22, SmallGroup(48,37)

Series: Derived Chief Lower central Upper central

C1C6 — C4○D12
C1C3C6D6C4×S3 — C4○D12
C3C6 — C4○D12
C1C4C2×C4

Generators and relations for C4○D12
 G = < a,b,c | a4=c2=1, b6=a2, ab=ba, ac=ca, cbc=a2b5 >

2C2
6C2
6C2
3C4
3C4
3C22
3C22
2C6
2S3
2S3
3C2×C4
3D4
3D4
3D4
3C2×C4
3Q8
3C4○D4

Character table of C4○D12

 class 12A2B2C2D34A4B4C4D4E6A6B6C12A12B12C12D
 size 112662112662222222
ρ1111111111111111111    trivial
ρ211-1-11111-1-111-1-11-1-11    linear of order 2
ρ3111-111-1-1-11-1111-1-1-1-1    linear of order 2
ρ41111-11-1-1-1-11111-1-1-1-1    linear of order 2
ρ511-11-1111-11-11-1-11-1-11    linear of order 2
ρ611-1111-1-11-1-11-1-1-111-1    linear of order 2
ρ711-1-1-11-1-11111-1-1-111-1    linear of order 2
ρ8111-1-11111-1-11111111    linear of order 2
ρ922-200-122-200-111-111-1    orthogonal lifted from D6
ρ1022200-1-2-2-200-1-1-11111    orthogonal lifted from D6
ρ1122-200-1-2-2200-1111-1-11    orthogonal lifted from D6
ρ1222200-122200-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ132-20002-2i2i000-200-2i002i    complex lifted from C4○D4
ρ142-200022i-2i000-2002i00-2i    complex lifted from C4○D4
ρ152-2000-12i-2i0001--3-3ζ2-33ζ2    complex faithful
ρ162-2000-12i-2i0001-3--3ζ23-3ζ2    complex faithful
ρ172-2000-1-2i2i0001--3-3ζ23-3ζ2    complex faithful
ρ182-2000-1-2i2i0001-3--3ζ2-33ζ2    complex faithful

Permutation representations of C4○D12
On 24 points - transitive group 24T19
Generators in S24
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)

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

G:=Group( (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22) );

G=PermutationGroup([(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,24),(11,23),(12,22)])

G:=TransitiveGroup(24,19);

On 24 points - transitive group 24T24
Generators in S24
(1 23 7 17)(2 24 8 18)(3 13 9 19)(4 14 10 20)(5 15 11 21)(6 16 12 22)
(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 20)(14 19)(15 18)(16 17)(21 24)(22 23)

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

G:=Group( (1,23,7,17)(2,24,8,18)(3,13,9,19)(4,14,10,20)(5,15,11,21)(6,16,12,22), (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)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23) );

G=PermutationGroup([(1,23,7,17),(2,24,8,18),(3,13,9,19),(4,14,10,20),(5,15,11,21),(6,16,12,22)], [(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),(13,20),(14,19),(15,18),(16,17),(21,24),(22,23)])

G:=TransitiveGroup(24,24);

Matrix representation of C4○D12 in GL2(𝔽13) generated by

80
08
,
610
33
,
121
01
G:=sub<GL(2,GF(13))| [8,0,0,8],[6,3,10,3],[12,0,1,1] >;

C4○D12 in GAP, Magma, Sage, TeX

C_4\circ D_{12}
% in TeX

G:=Group("C4oD12");
// GroupNames label

G:=SmallGroup(48,37);
// by ID

G=gap.SmallGroup(48,37);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,46,182,804]);
// Polycyclic

G:=Group<a,b,c|a^4=c^2=1,b^6=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^5>;
// generators/relations

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