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G = C42.13D4order 128 = 27

13rd non-split extension by C42 of D4 acting faithfully

p-group, non-abelian, nilpotent (class 4), monomial

Aliases: C42.13D4, 2+ 1+4.2C22, (C2×D4).35D4, C22⋊C4.1D4, C42⋊C46C2, C2.22C2≀C22, D44D4.2C2, (C2×D4).3C23, (C22×C4).26D4, C23.15(C2×D4), C23.D42C2, C23.7D42C2, C23⋊C4.2C22, C22.46C22≀C2, C41D4.56C22, C4.D4.2C22, C22.53C242C2, C22.D4.4C22, (C2×C4).15(C2×D4), SmallGroup(128,930)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×D4 — C42.13D4
C1C2C22C23C2×D4C22.D4C22.53C24 — C42.13D4
C1C2C22C2×D4 — C42.13D4
C1C2C22C2×D4 — C42.13D4
C1C2C22C2×D4 — C42.13D4

Generators and relations for C42.13D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=a-1b, cbc-1=a2b, bd=db, dcd=b2c-1 >

Subgroups: 336 in 123 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2 [×5], C4 [×10], C22, C22 [×10], C8, C2×C4, C2×C4 [×13], D4 [×10], Q8 [×3], C23 [×2], C23 [×3], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×5], M4(2), D8, SD16, C22×C4 [×2], C22×C4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C4○D4 [×2], C23⋊C4 [×2], C23⋊C4 [×2], C4.D4, C4≀C2, C4×D4 [×2], C4×Q8, C22.D4 [×2], C22.D4 [×3], C4.4D4 [×2], C41D4, C8⋊C22, 2+ 1+4, C23.D4 [×2], C42⋊C4, D44D4, C23.7D4 [×2], C22.53C24, C42.13D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C2≀C22, C42.13D4

Character table of C42.13D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L8
 size 11244884444444888161616
ρ111111111111111111111    trivial
ρ2111111-1-1-1-11-11111-11-1-1    linear of order 2
ρ3111111-1-1-11-11-11-111-11-1    linear of order 2
ρ4111111111-1-1-1-11-11-1-1-11    linear of order 2
ρ511111-1111-1-1-1-11-1-1-111-1    linear of order 2
ρ611111-1-1-1-11-11-11-1-111-11    linear of order 2
ρ711111-1-1-1-1-11-1111-1-1-111    linear of order 2
ρ811111-1111111111-11-1-1-1    linear of order 2
ρ9222-220000-20-20-2002000    orthogonal lifted from D4
ρ10222-2200002020-200-2000    orthogonal lifted from D4
ρ11222-2-20-22200002000000    orthogonal lifted from D4
ρ122222-200000202-2-200000    orthogonal lifted from D4
ρ132222-200000-20-2-2200000    orthogonal lifted from D4
ρ14222-2-202-2-200002000000    orthogonal lifted from D4
ρ1544-400-200000000020000    orthogonal lifted from C2≀C22
ρ1644-4002000000000-20000    orthogonal lifted from C2≀C22
ρ174-400000-22-2i2i2i-2i0000000    complex faithful
ρ184-4000002-22i2i-2i-2i0000000    complex faithful
ρ194-4000002-2-2i-2i2i2i0000000    complex faithful
ρ204-400000-222i-2i-2i2i0000000    complex faithful

Permutation representations of C42.13D4
On 16 points - transitive group 16T344
Generators in S16
(9 10 11 12)(13 14 15 16)
(1 2 3 4)(5 6 8 7)(9 10 11 12)(13 14 15 16)
(1 13 3 15)(2 14)(4 16)(5 12 8 10)(6 9)(7 11)
(1 12)(2 9)(3 10)(4 11)(5 15)(6 16)(7 14)(8 13)

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

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

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

G:=TransitiveGroup(16,344);

On 16 points - transitive group 16T380
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 8 2 7)(3 6 4 5)(9 12 11 10)(13 16 15 14)
(1 16 6 10)(2 14 5 12)(3 11 8 15)(4 9 7 13)
(1 13)(2 15)(3 10)(4 12)(5 11)(6 9)(7 14)(8 16)

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

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

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

G:=TransitiveGroup(16,380);

Matrix representation of C42.13D4 in GL4(𝔽5) generated by

0212
0100
0003
2101
,
4004
2021
0203
2001
,
2002
0043
0004
0203
,
1020
0031
0040
0130
G:=sub<GL(4,GF(5))| [0,0,0,2,2,1,0,1,1,0,0,0,2,0,3,1],[4,2,0,2,0,0,2,0,0,2,0,0,4,1,3,1],[2,0,0,0,0,0,0,2,0,4,0,0,2,3,4,3],[1,0,0,0,0,0,0,1,2,3,4,3,0,1,0,0] >;

C42.13D4 in GAP, Magma, Sage, TeX

C_4^2._{13}D_4
% in TeX

G:=Group("C4^2.13D4");
// GroupNames label

G:=SmallGroup(128,930);
// by ID

G=gap.SmallGroup(128,930);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,456,422,352,297,1971,375,4037]);
// Polycyclic

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

Export

Character table of C42.13D4 in TeX

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