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G = C42.4D4order 128 = 27

4th non-split extension by C42 of D4 acting faithfully

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

Aliases: C42.4D4, 2- 1+4.C4, C2.11C2≀C4, (C2×Q8).2D4, C4.10D4.C4, C4⋊Q8.1C22, C4.6Q161C2, C42.3C44C2, D4.10D4.1C2, C22.4(C23⋊C4), (C2×Q8).2(C2×C4), (C2×C4).8(C22⋊C4), SmallGroup(128,137)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×Q8 — C42.4D4
C1C2C22C2×C4C2×Q8C4⋊Q8D4.10D4 — C42.4D4
C1C2C22C2×C4C2×Q8 — C42.4D4
C1C2C22C2×C4C4⋊Q8 — C42.4D4
C1C2C2C22C2×C4C4⋊Q8 — C42.4D4

Generators and relations for C42.4D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a-1b2, ab=ba, cac-1=a-1b, ad=da, cbc-1=a2b, dbd-1=a2b-1, dcd-1=a-1c3 >

2C2
8C2
2C4
2C4
2C4
4C4
4C4
4C4
4C22
2C2×C4
2D4
2C2×C4
2C2×C4
2Q8
4Q8
4D4
4C2×C4
4Q8
4C8
4C2×C4
4Q8
4C2×C4
4D4
8C8
8C8
2C4○D4
2M4(2)
4C4○D4
4C2×Q8
4Q16
4C2×C8
4C4○D4
4SD16
4M4(2)
4C4⋊C4
2C4⋊C8
2C4≀C2
2C8.C22
2C4.10D4

Character table of C42.4D4

 class 12A2B2C4A4B4C4D4E4F8A8B8C8D8E8F8G
 size 11284448888888161616
ρ111111111111111111    trivial
ρ2111-1111-111-1-1-1-111-1    linear of order 2
ρ31111111111-1-1-1-1-1-11    linear of order 2
ρ4111-1111-1111111-1-1-1    linear of order 2
ρ5111-1-11-1-11-1-iii-ii-i1    linear of order 4
ρ61111-11-111-1i-i-iii-i-1    linear of order 4
ρ71111-11-111-1-iii-i-ii-1    linear of order 4
ρ8111-1-11-1-11-1i-i-ii-ii1    linear of order 4
ρ92220-22-20-220000000    orthogonal lifted from D4
ρ1022202220-2-20000000    orthogonal lifted from D4
ρ1144-4-20002000000000    orthogonal lifted from C2≀C4
ρ1244400-400000000000    orthogonal lifted from C23⋊C4
ρ1344-42000-2000000000    orthogonal lifted from C2≀C4
ρ144-400-2020002-22-2000    symplectic faithful, Schur index 2
ρ154-400-202000-22-22000    symplectic faithful, Schur index 2
ρ164-40020-2000--2--2-2-2000    complex faithful
ρ174-40020-2000-2-2--2--2000    complex faithful

Permutation representations of C42.4D4
On 16 points - transitive group 16T361
Generators in S16
(1 5)(2 11 6 15)(3 7)(4 9 8 13)(10 14)(12 16)
(1 14 5 10)(2 15 6 11)(3 12 7 16)(4 13 8 9)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 13 11 4 6 9 15 8)(3 12)(7 16)(10 14)

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

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

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

G:=TransitiveGroup(16,361);

Matrix representation of C42.4D4 in GL4(𝔽3) generated by

2000
0020
0100
0002
,
0002
0010
0200
1000
,
0210
0001
1000
0220
,
2000
0220
0120
0001
G:=sub<GL(4,GF(3))| [2,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2],[0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,0],[0,0,1,0,2,0,0,2,1,0,0,2,0,1,0,0],[2,0,0,0,0,2,1,0,0,2,2,0,0,0,0,1] >;

C42.4D4 in GAP, Magma, Sage, TeX

C_4^2._4D_4
% in TeX

G:=Group("C4^2.4D4");
// GroupNames label

G:=SmallGroup(128,137);
// by ID

G=gap.SmallGroup(128,137);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,723,352,346,745,248,1684,1411,718,375,172,4037,2028]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^-1*b^2,a*b=b*a,c*a*c^-1=a^-1*b,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^-1*c^3>;
// generators/relations

Export

Subgroup lattice of C42.4D4 in TeX
Character table of C42.4D4 in TeX

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