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G = C12.46D4  order 96 = 25·3

3rd non-split extension by C12 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.46D4, C4.11D12, M4(2)⋊3S3, (C2×C4).1D6, (C22×S3).C4, C2.9(D6⋊C4), (C2×D12).6C2, C3⋊1(C4.D4), C4.Dic3⋊2C2, C22.4(C4×S3), C4.21(C3⋊D4), C6.8(C22⋊C4), (C3×M4(2))⋊7C2, (C2×C12).13C22, (C2×C6).2(C2×C4), SmallGroup(96,30)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C12.46D4
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C12.46D4
Lower central
C3 — C6 — C2×C6 — C12.46D4
Upper central
C1 — C2 — C2×C4 — M4(2)

Generators and relations for C12.46D4
 G = < a,b,c,d | a8=b2=c3=d2=1, bab=a5, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

C1
C2
2C2
12C2
12C2
C3
C4
C22
C4
6C22
6C22
12C22
12C22
C6
2C6
4S3
4S3
C2×C4
2C8
3C23
3C23
6C8
6D4
6D4
C2×C6
C12
C12
2D6
2D6
4D6
4D6
M4(2)
3M4(2)
3C2×D4
C22×S3
C22×S3
C2×C12
2C3⋊C8
2C24
2D12
2D12
3C4.D4
C3×M4(2)
C2×D12
C4.Dic3
C12.46D4

Character table of C12.46D4

 class 12A2B2C2D34A4B6A6B8A8B8C8D12A12B12C24A24B24C24D
 size 1121212222244412122244444
ρ1111111111111111111111    trivial
ρ2111-1-11111111-1-11111111    linear of order 2
ρ3111-1-111111-1-111111-1-1-1-1    linear of order 2
ρ41111111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ5111-111-1-111-iii-i-1-1-1ii-i-i    linear of order 4
ρ61111-11-1-111-ii-ii-1-1-1ii-i-i    linear of order 4
ρ7111-111-1-111i-i-ii-1-1-1-i-iii    linear of order 4
ρ81111-11-1-111i-ii-i-1-1-1-i-iii    linear of order 4
ρ922200-122-1-1-2-200-1-1-11111    orthogonal lifted from D6
ρ1022-20022-22-2000022-20000    orthogonal lifted from D4
ρ1122200-122-1-12200-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1222-2002-222-20000-2-220000    orthogonal lifted from D4
ρ1322-200-1-22-11000011-1-√3√3√3-√3    orthogonal lifted from D12
ρ1422-200-1-22-11000011-1√3-√3-√3√3    orthogonal lifted from D12
ρ1522200-1-2-2-1-1-2i2i00111-i-iii    complex lifted from C4×S3
ρ1622200-1-2-2-1-12i-2i00111ii-i-i    complex lifted from C4×S3
ρ1722-200-12-2-110000-1-11-√-3√-3-√-3√-3    complex lifted from C3⋊D4
ρ1822-200-12-2-110000-1-11√-3-√-3√-3-√-3    complex lifted from C3⋊D4
ρ194-4000400-4000000000000    orthogonal lifted from C4.D4
ρ204-4000-200200000-2√32√300000    orthogonal faithful
ρ214-4000-2002000002√3-2√300000    orthogonal faithful

Permutation representations of C12.46D4
►On 24 points - transitive group 24T105
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)

(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)

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

(2 6)(3 7)(9 24)(10 21)(11 22)(12 19)(13 20)(14 17)(15 18)(16 23)
 
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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23), (1,11,22)(2,12,23)(3,13,24)(4,14,17)(5,15,18)(6,16,19)(7,9,20)(8,10,21), (2,6)(3,7)(9,24)(10,21)(11,22)(12,19)(13,20)(14,17)(15,18)(16,23)>;
 
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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23), (1,11,22)(2,12,23)(3,13,24)(4,14,17)(5,15,18)(6,16,19)(7,9,20)(8,10,21), (2,6)(3,7)(9,24)(10,21)(11,22)(12,19)(13,20)(14,17)(15,18)(16,23) );
 
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)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23)], [(1,11,22),(2,12,23),(3,13,24),(4,14,17),(5,15,18),(6,16,19),(7,9,20),(8,10,21)], [(2,6),(3,7),(9,24),(10,21),(11,22),(12,19),(13,20),(14,17),(15,18),(16,23)]])
 
G:=TransitiveGroup(24,105);
 

C12.46D4 is a maximal subgroup of
 S3×C4.D4  D12.3D4  M4(2).21D6  D12.6D4  Q8⋊5D12  C42⋊5D6  C24.19D4  C24.42D4  M4(2).31D6  Q8.8D12  Q8.9D12  D12⋊18D4  M4(2).D6  D12.39D4  M4(2).15D6  C36.48D4  C12.D12  C12.70D12  C12.19D12  C20.5D12  C60.29D4  M4(2)⋊D15  Dic5.D12
C12.46D4 is a maximal quotient of
 C42.D6  (C22×S3)⋊C8  C4.Dic12  C4.D24  M4(2)⋊Dic3  C36.48D4  C12.D12  C12.70D12  C12.19D12  C20.5D12  C60.29D4  M4(2)⋊D15  Dic5.D12

Matrix representation of C12.46D4 ►in GL6(𝔽73)

010000
100000
000010
001168213
0007200
00432375
,
7200000
0720000
001000
000100
0000720
001752072
,
36280000
28360000
001000
000100
000010
000001
,
100000
0720000
000100
001000
001168213
002274852

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,43,0,0,0,68,72,23,0,0,1,21,0,7,0,0,0,3,0,5],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,17,0,0,0,1,0,52,0,0,0,0,72,0,0,0,0,0,0,72],[36,28,0,0,0,0,28,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,11,22,0,0,1,0,68,7,0,0,0,0,21,48,0,0,0,0,3,52] >;
 

C12.46D4 in GAP, Magma, Sage, TeX 

C_{12}._{46}D_4
 
% in TeX
 
G:=Group("C12.46D4");
 
// GroupNames label
 
G:=SmallGroup(96,30);
 
// by ID
 
G=gap.SmallGroup(96,30);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,362,86,297,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^3=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C12.46D4 in TeX
Character table of C12.46D4 in TeX

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