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G = C42.D4order 128 = 27

1st non-split extension by C42 of D4 acting faithfully

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

Aliases: C42.1D4, 2+ 1+4⋊C4, C2.8C2≀C4, C4.D4⋊C4, (C2×D4).1D4, C4.D81C2, C42⋊C41C2, D44D4.1C2, C41D4.1C22, C22.1(C23⋊C4), (C2×D4).1(C2×C4), (C2×C4).5(C22⋊C4), SmallGroup(128,134)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×D4 — C42.D4
C1C2C22C2×C4C2×D4C41D4D44D4 — C42.D4
C1C2C22C2×C4C2×D4 — C42.D4
C1C2C22C2×C4C41D4 — C42.D4
C1C2C2C22C2×C4C41D4 — C42.D4

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

Subgroups: 232 in 61 conjugacy classes, 14 normal (all characteristic)
C1, C2, C2 [×4], C4 [×5], C22, C22 [×6], C8 [×2], C2×C4, C2×C4 [×4], D4 [×8], Q8, C23 [×3], C42, C22⋊C4, C2×C8, M4(2), D8, SD16, C2×D4 [×2], C2×D4 [×3], C4○D4 [×2], C23⋊C4, C4.D4, C4≀C2, C4⋊C8, C41D4, C8⋊C22, 2+ 1+4, C4.D8, C42⋊C4, D44D4, C42.D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C23⋊C4, C2≀C4, C42.D4

Character table of C42.D4

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E
 size 11288844481616888816
ρ111111111111111111    trivial
ρ21111-11111-111-1-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-1-1-11    linear of order 2
ρ41111-11111-1-1-11111-1    linear of order 2
ρ51111-1-1-1-11-1-iii-i-ii1    linear of order 4
ρ611111-1-1-111-ii-iii-i-1    linear of order 4
ρ71111-1-1-1-11-1i-i-iii-i1    linear of order 4
ρ811111-1-1-111i-ii-i-ii-1    linear of order 4
ρ9222-202-2-2200000000    orthogonal lifted from D4
ρ10222-20-222200000000    orthogonal lifted from D4
ρ1144-40-2000020000000    orthogonal lifted from C2≀C4
ρ1244400000-400000000    orthogonal lifted from C23⋊C4
ρ1344-4020000-20000000    orthogonal lifted from C2≀C4
ρ144-40000-220000-22-220    orthogonal faithful
ρ154-40000-2200002-22-20    orthogonal faithful
ρ164-400002-20000-2-2--2--20    complex faithful
ρ174-400002-20000--2--2-2-20    complex faithful

Permutation representations of C42.D4
On 16 points - transitive group 16T335
Generators in S16
(1 3)(2 4)(5 7)(6 8)(9 15 13 11)(10 16 14 12)
(1 6 3 8)(2 5 4 7)(9 11 13 15)(10 12 14 16)
(1 14)(2 11 5 13)(3 10)(4 15 7 9)(6 16 8 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,335);

On 16 points - transitive group 16T343
Generators in S16
(9 15 13 11)(10 16 14 12)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 16 14 12)
(1 10 3 16)(2 9)(4 11 8 15)(5 14 7 12)(6 13)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,343);

On 16 points - transitive group 16T387
Generators in S16
(9 15 13 11)(10 16 14 12)
(1 4 2 3)(5 8 7 6)(9 15 13 11)(10 16 14 12)
(1 9 7 16)(2 13 5 12)(3 11 6 14)(4 15 8 10)
(3 4)(5 6)(7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,387);

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

1210
2001
2200
2100
,
1001
0111
2120
1002
,
0012
1112
0200
0022
,
0001
1001
0100
0011
G:=sub<GL(4,GF(3))| [1,2,2,2,2,0,2,1,1,0,0,0,0,1,0,0],[1,0,2,1,0,1,1,0,0,1,2,0,1,1,0,2],[0,1,0,0,0,1,2,0,1,1,0,2,2,2,0,2],[0,1,0,0,0,0,1,0,0,0,0,1,1,1,0,1] >;

C42.D4 in GAP, Magma, Sage, TeX

C_4^2.D_4
% in TeX

G:=Group("C4^2.D4");
// GroupNames label

G:=SmallGroup(128,134);
// by ID

G=gap.SmallGroup(128,134);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,1242,745,1684,1411,718,375,172,4037,2028]);
// 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,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^-1>;
// generators/relations

Export

Character table of C42.D4 in TeX

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