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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--3-3-i-33i    complex faithful
ρ162-2000-12i-2i0001-3--3-i3-3i    complex faithful
ρ172-2000-1-2i2i0001--3-3i3-3-i    complex faithful
ρ182-2000-1-2i2i0001-3--3i-33-i    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 16)(2 15)(3 14)(4 13)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)

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

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

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

G:=TransitiveGroup(24,19);

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

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

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

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

G:=TransitiveGroup(24,24);

C4○D12 is a maximal subgroup of
C424S3  D12⋊C4  C8○D12  C4○D24  D12.C4  C8⋊D6  C8.D6  D126C22  Q8.11D6  Q8.13D6  D46D6  Q8.15D6  S3×C4○D4  D4○D12  Q8○D12  D365C2  D125S3  D6.D6  D6.6D6  D6.3D6  C12.59D6  C24.10D6  GL2(𝔽3)⋊C22  D6.D10  D125D5  C12.28D10  Dic3.D10  D6011C2  D6.D14  D125D7  D14.D6  Dic3.D14  D8411C2  C4.6S5
C4○D12 is a maximal quotient of
C4×Dic6  C12.6Q8  C422S3  C4×D12  C427S3  C423S3  C23.8D6  C23.9D6  Dic3⋊D4  C23.11D6  Dic3.Q8  D6.D4  D6⋊Q8  C4⋊C4⋊S3  C12.48D4  C23.26D6  C4×C3⋊D4  C23.28D6  C127D4  D365C2  D125S3  D6.D6  D6.6D6  D6.3D6  C12.59D6  C24.10D6  D6.D10  D125D5  C12.28D10  Dic3.D10  D6011C2  D6.D14  D125D7  D14.D6  Dic3.D14  D8411C2

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

Export

Subgroup lattice of C4○D12 in TeX
Character table of C4○D12 in TeX

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