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G = C42.17D4order 128 = 27

17th non-split extension by C42 of D4 acting faithfully

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

Aliases: C42.17D4, 2- 1+4.2C22, (C2×D4).38D4, (C2×Q8).37D4, C2.28C2≀C22, D4.8D42C2, C4⋊Q8.98C22, D4.10D42C2, (C2×Q8).3C23, C42.3C42C2, C42.C43C2, C22.52C22≀C2, C4.10D4.2C22, C4.4D4.23C22, C22.49C242C2, (C2×C4).21(C2×D4), SmallGroup(128,936)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×Q8 — C42.17D4
C1C2C22C2×C4C2×Q8C4.4D4C22.49C24 — C42.17D4
C1C2C22C2×Q8 — C42.17D4
C1C2C22C2×Q8 — C42.17D4
C1C2C22C2×Q8 — C42.17D4

Generators and relations for C42.17D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1b, cbc-1=a2b, bd=db, dcd=b2c3 >

Subgroups: 304 in 118 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2 [×4], C4 [×9], C22, C22 [×7], C8 [×3], C2×C4, C2×C4 [×2], C2×C4 [×11], D4 [×9], Q8 [×5], C23 [×2], C42, C42 [×2], C42, C22⋊C4 [×6], C4⋊C4 [×3], M4(2) [×3], D8 [×2], SD16 [×3], Q16, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8, C4○D4 [×4], C4.10D4, C4.10D4 [×2], C4≀C2 [×3], C42⋊C2 [×2], C4×D4, C4⋊D4 [×2], C4.4D4 [×2], C4.4D4, C4⋊Q8, C8⋊C22 [×2], C8.C22, 2- 1+4, C42.C4 [×2], C42.3C4, D4.8D4 [×2], D4.10D4, C22.49C24, C42.17D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C2≀C22, C42.17D4

Character table of C42.17D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K8A8B8C
 size 11288844444444488161616
ρ111111111111111111111    trivial
ρ21111-1-11-1-1111-1-11111-1-1    linear of order 2
ρ3111-1-1-11-1-1111-1-11-11-111    linear of order 2
ρ4111-111111111111-11-1-1-1    linear of order 2
ρ5111-1-11-11-1-1111-11-1-11-11    linear of order 2
ρ6111-11-1-1-11-111-111-1-111-1    linear of order 2
ρ711111-1-1-11-111-1111-1-1-11    linear of order 2
ρ81111-11-11-1-1111-111-1-11-1    linear of order 2
ρ92220020-2002-2-20-200000    orthogonal lifted from D4
ρ1022200-202002-220-200000    orthogonal lifted from D4
ρ112220002002-2-20020-2000    orthogonal lifted from D4
ρ12222000-200-2-2-200202000    orthogonal lifted from D4
ρ1322202000-20-220-2-200000    orthogonal lifted from D4
ρ142220-200020-2202-200000    orthogonal lifted from D4
ρ1544-4-20000000000020000    orthogonal lifted from C2≀C22
ρ1644-4200000000000-20000    orthogonal lifted from C2≀C22
ρ174-40000-22i-2i200-2i2i000000    complex faithful
ρ184-40000-2-2i2i2002i-2i000000    complex faithful
ρ194-400002-2i-2i-2002i2i000000    complex faithful
ρ204-4000022i2i-200-2i-2i000000    complex faithful

Permutation representations of C42.17D4
On 16 points - transitive group 16T337
Generators in S16
(1 9 5 13)(3 15 7 11)
(1 9 5 13)(2 14 6 10)(3 15 7 11)(4 12 8 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 6)(2 5)(3 4)(7 8)(9 10)(11 16)(12 15)(13 14)

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

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

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

G:=TransitiveGroup(16,337);

On 16 points - transitive group 16T407
Generators in S16
(1 14 5 10)(3 12 7 16)
(1 14 5 10)(2 11 6 15)(3 12 7 16)(4 9 8 13)
(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 10)(11 16)(12 15)(13 14)

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

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

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

G:=TransitiveGroup(16,407);

Matrix representation of C42.17D4 in GL4(𝔽5) generated by

0004
0400
0040
1000
,
0001
0020
0200
4000
,
0040
1000
0002
0300
,
0400
4000
0002
0030
G:=sub<GL(4,GF(5))| [0,0,0,1,0,4,0,0,0,0,4,0,4,0,0,0],[0,0,0,4,0,0,2,0,0,2,0,0,1,0,0,0],[0,1,0,0,0,0,0,3,4,0,0,0,0,0,2,0],[0,4,0,0,4,0,0,0,0,0,0,3,0,0,2,0] >;

C42.17D4 in GAP, Magma, Sage, TeX

C_4^2._{17}D_4
% in TeX

G:=Group("C4^2.17D4");
// GroupNames label

G:=SmallGroup(128,936);
// by ID

G=gap.SmallGroup(128,936);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,723,2019,1018,297,248,2804,1971,718,375,172,4037]);
// Polycyclic

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

Export

Character table of C42.17D4 in TeX

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