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G = D12⋊C4order 96 = 25·3

4th semidirect product of D12 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D124C4, Dic64C4, C12.54D4, M4(2)⋊4S3, C22.3D12, C32C4≀C2, C4.3(C4×S3), (C2×C6).1D4, C12.6(C2×C4), (C2×C4).38D6, C4○D12.2C2, (C4×Dic3)⋊1C2, C2.11(D6⋊C4), C4.29(C3⋊D4), (C3×M4(2))⋊8C2, C6.10(C22⋊C4), (C2×C12).15C22, SmallGroup(96,32)

Series: Derived Chief Lower central Upper central

C1C12 — D12⋊C4
C1C3C6C12C2×C12C4○D12 — D12⋊C4
C3C6C12 — D12⋊C4
C1C4C2×C4M4(2)

Generators and relations for D12⋊C4
 G = < a,b,c | a12=b2=c4=1, bab=a-1, cac-1=a5, cbc-1=a7b >

2C2
12C2
6C4
6C22
6C4
6C4
2C6
4S3
2C8
3D4
3Q8
6C2×C4
6C2×C4
6D4
2Dic3
2Dic3
2D6
2Dic3
3C42
3C4○D4
2C4×S3
2C3⋊D4
2C2×Dic3
2C24
3C4≀C2

Character table of D12⋊C4

 class 12A2B2C34A4B4C4D4E4F4G4H6A6B8A8B12A12B12C24A24B24C24D
 size 11212211266661224442244444
ρ1111111111111111111111111    trivial
ρ211111111-1-1-1-1111-1-1111-1-1-1-1    linear of order 2
ρ3111-11111-1-1-1-1-111111111111    linear of order 2
ρ4111-111111111-111-1-1111-1-1-1-1    linear of order 2
ρ511-111-1-11i-ii-i-11-1-ii-1-11i-i-ii    linear of order 4
ρ611-1-11-1-11-ii-ii11-1-ii-1-11i-i-ii    linear of order 4
ρ711-111-1-11-ii-ii-11-1i-i-1-11-iii-i    linear of order 4
ρ811-1-11-1-11i-ii-i11-1i-i-1-11-iii-i    linear of order 4
ρ92220-122200000-1-122-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ102220-122200000-1-1-2-2-1-1-11111    orthogonal lifted from D6
ρ1122202-2-2-2000002200-2-2-20000    orthogonal lifted from D4
ρ1222-20222-2000002-20022-20000    orthogonal lifted from D4
ρ132220-1-2-2-200000-1-100111-33-33    orthogonal lifted from D12
ρ142220-1-2-2-200000-1-1001113-33-3    orthogonal lifted from D12
ρ152-2002-2i2i0-1+i1+i1-i-1-i0-2000-2i2i00000    complex lifted from C4≀C2
ρ162-20022i-2i0-1-i1-i1+i-1+i0-20002i-2i00000    complex lifted from C4≀C2
ρ1722-20-122-200000-1100-1-11--3--3-3-3    complex lifted from C3⋊D4
ρ1822-20-122-200000-1100-1-11-3-3--3--3    complex lifted from C3⋊D4
ρ192-20022i-2i01+i-1+i-1-i1-i0-20002i-2i00000    complex lifted from C4≀C2
ρ2022-20-1-2-2200000-11-2i2i11-1-iii-i    complex lifted from C4×S3
ρ2122-20-1-2-2200000-112i-2i11-1i-i-ii    complex lifted from C4×S3
ρ222-2002-2i2i01-i-1-i-1+i1+i0-2000-2i2i00000    complex lifted from C4≀C2
ρ234-400-24i-4i0000002000-2i2i00000    complex faithful
ρ244-400-2-4i4i00000020002i-2i00000    complex faithful

Permutation representations of D12⋊C4
On 24 points - transitive group 24T106
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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
(2 6)(3 11)(5 9)(8 12)(13 16 19 22)(14 21 20 15)(17 24 23 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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13), (2,6)(3,11)(5,9)(8,12)(13,16,19,22)(14,21,20,15)(17,24,23,18)>;

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

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

G:=TransitiveGroup(24,106);

D12⋊C4 is a maximal subgroup of
M4(2)⋊D6  D121D4  D12.4D4  D12.5D4  S3×C4≀C2  C423D6  D2410C4  D247C4  M4(2)⋊24D6  C24.100D4  C24.54D4  D1218D4  D12.38D4  D12.39D4  D12.40D4  Dic18⋊C4  D124Dic3  C12.80D12  C62.37D4  C60.98D4  D6013C4  D6010C4  D124F5  D602C4
D12⋊C4 is a maximal quotient of
C42.D6  C42.2D6  C23.35D12  C22.2D24  D122C8  Dic62C8  C12.3C42  Dic18⋊C4  D124Dic3  C12.80D12  C62.37D4  C60.98D4  D6013C4  D6010C4  D124F5  D602C4

Matrix representation of D12⋊C4 in GL4(𝔽5) generated by

3001
0240
0400
1000
,
0300
2000
4003
0120
,
2004
0100
0240
0003
G:=sub<GL(4,GF(5))| [3,0,0,1,0,2,4,0,0,4,0,0,1,0,0,0],[0,2,4,0,3,0,0,1,0,0,0,2,0,0,3,0],[2,0,0,0,0,1,2,0,0,0,4,0,4,0,0,3] >;

D12⋊C4 in GAP, Magma, Sage, TeX

D_{12}\rtimes C_4
% in TeX

G:=Group("D12:C4");
// GroupNames label

G:=SmallGroup(96,32);
// by ID

G=gap.SmallGroup(96,32);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,86,579,297,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^12=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^7*b>;
// generators/relations

Export

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

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