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G = C42.3D4order 128 = 27

3rd non-split extension by C42 of D4 acting faithfully

p-group, non-abelian, nilpotent (class 5), monomial

Aliases: C42.3D4, 2- 1+4⋊C4, C2.10C2≀C4, C4.10D4⋊C4, (C2×D4).2D4, C423C41C2, D4.8D4.1C2, C22.3(C23⋊C4), C4.4D4.2C22, C42.C226C2, (C2×Q8).1(C2×C4), (C2×C4).7(C22⋊C4), SmallGroup(128,136)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×Q8 — C42.3D4
C1C2C22C2×C4C2×D4C4.4D4D4.8D4 — C42.3D4
C1C2C22C2×C4C2×Q8 — C42.3D4
C1C2C22C2×C4C4.4D4 — C42.3D4
C1C2C2C22C2×C4C4.4D4 — C42.3D4

Generators and relations for C42.3D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a-1, ab=ba, cac-1=a-1b-1, ad=da, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a-1c-1 >

2C2
8C2
8C2
2C4
4C4
4C22
4C22
4C4
4C4
8C22
16C4
2Q8
2C2×C4
2C23
2D4
2C2×C4
4C8
4D4
4C2×C4
4C8
4C2×C4
4Q8
4D4
4D4
4C8
4C2×C4
4Q8
8C2×C4
2M4(2)
2C4○D4
4C2×Q8
4C4○D4
4C4○D4
4D8
4SD16
4C2×C8
4C22⋊C4
4C22⋊C4
2C8⋊C4
2C4≀C2
2C8⋊C22
2C23⋊C4

Character table of C42.3D4

 class 12A2B2C2D4A4B4C4D4E4F4G8A8B8C8D8E
 size 11288444881616888816
ρ111111111111111111    trivial
ρ2111-11111-11-1-11111-1    linear of order 2
ρ3111-11111-1111-1-1-1-1-1    linear of order 2
ρ41111111111-1-1-1-1-1-11    linear of order 2
ρ5111-1-1-1-11-11i-i-iii-i1    linear of order 4
ρ61111-1-1-1111-ii-iii-i-1    linear of order 4
ρ71111-1-1-1111i-ii-i-ii-1    linear of order 4
ρ8111-1-1-1-11-11-iii-i-ii1    linear of order 4
ρ92220-22220-20000000    orthogonal lifted from D4
ρ1022202-2-220-20000000    orthogonal lifted from D4
ρ1144-4-20000200000000    orthogonal lifted from C2≀C4
ρ124440000-4000000000    orthogonal lifted from C23⋊C4
ρ1344-420000-200000000    orthogonal lifted from C2≀C4
ρ144-4000-2i2i000001-i-1-i1+i-1+i0    complex faithful
ρ154-4000-2i2i00000-1+i1+i-1-i1-i0    complex faithful
ρ164-40002i-2i00000-1-i1-i-1+i1+i0    complex faithful
ρ174-40002i-2i000001+i-1+i1-i-1-i0    complex faithful

Permutation representations of C42.3D4
On 16 points - transitive group 16T397
Generators in S16
(5 7)(6 8)(9 15 13 11)(10 16 14 12)
(1 4 2 3)(5 8 7 6)(9 15 13 11)(10 12 14 16)
(1 13 5 16)(2 9 7 12)(3 11 8 14)(4 15 6 10)
(1 2)(3 4)(5 6 7 8)(9 10 11 12 13 14 15 16)

G:=sub<Sym(16)| (5,7)(6,8)(9,15,13,11)(10,16,14,12), (1,4,2,3)(5,8,7,6)(9,15,13,11)(10,12,14,16), (1,13,5,16)(2,9,7,12)(3,11,8,14)(4,15,6,10), (1,2)(3,4)(5,6,7,8)(9,10,11,12,13,14,15,16)>;

G:=Group( (5,7)(6,8)(9,15,13,11)(10,16,14,12), (1,4,2,3)(5,8,7,6)(9,15,13,11)(10,12,14,16), (1,13,5,16)(2,9,7,12)(3,11,8,14)(4,15,6,10), (1,2)(3,4)(5,6,7,8)(9,10,11,12,13,14,15,16) );

G=PermutationGroup([(5,7),(6,8),(9,15,13,11),(10,16,14,12)], [(1,4,2,3),(5,8,7,6),(9,15,13,11),(10,12,14,16)], [(1,13,5,16),(2,9,7,12),(3,11,8,14),(4,15,6,10)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12,13,14,15,16)])

G:=TransitiveGroup(16,397);

Matrix representation of C42.3D4 in GL4(𝔽5) generated by

1313
3321
1402
4120
,
1040
1330
2040
3032
,
0032
1011
0122
0003
,
0410
0314
0433
1322
G:=sub<GL(4,GF(5))| [1,3,1,4,3,3,4,1,1,2,0,2,3,1,2,0],[1,1,2,3,0,3,0,0,4,3,4,3,0,0,0,2],[0,1,0,0,0,0,1,0,3,1,2,0,2,1,2,3],[0,0,0,1,4,3,4,3,1,1,3,2,0,4,3,2] >;

C42.3D4 in GAP, Magma, Sage, TeX

C_4^2._3D_4
% in TeX

G:=Group("C4^2.3D4");
// GroupNames label

G:=SmallGroup(128,136);
// by ID

G=gap.SmallGroup(128,136);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,723,352,346,745,1684,1411,718,375,172,4037,2028]);
// Polycyclic

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

Export

Subgroup lattice of C42.3D4 in TeX
Character table of C42.3D4 in TeX

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