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G = C42.8D4order 128 = 27

8th non-split extension by C42 of D4 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.8D4, (C2×Q8)⋊3Q8, (C2×D4)⋊3Q8, C22⋊C4.5D4, C4.9C4213C2, C426C431C2, C23.140(C2×D4), C4.99(C22⋊Q8), C22.42C22≀C2, C4.99(C4.4D4), M4(2)⋊4C419C2, C22.3(C22⋊Q8), (C22×C4).721C23, (C2×C42).359C22, C42⋊C22.8C2, C2.12(C23⋊Q8), C22.8(C4.4D4), C42⋊C2.55C22, C23.37C231C2, C23.C23.8C2, (C2×M4(2)).222C22, (C2×C4≀C2).15C2, (C2×C4).17(C2×Q8), (C2×C4).1370(C2×D4), (C2×C4).769(C4○D4), (C2×C4○D4).56C22, SmallGroup(128,763)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C42.8D4
C1C2C4C2×C4C22×C4C2×C4○D4C2×C4≀C2 — C42.8D4
C1C2C22×C4 — C42.8D4
C1C4C22×C4 — C42.8D4
C1C2C2C22×C4 — C42.8D4

Generators and relations for C42.8D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=a-1b-1, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2b2c3 >

Subgroups: 264 in 122 conjugacy classes, 40 normal (36 characteristic)
C1, C2, C2 [×4], C4 [×4], C4 [×9], C22 [×3], C22 [×3], C8 [×2], C2×C4 [×6], C2×C4 [×12], D4 [×4], Q8 [×6], C23, C23, C42 [×2], C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C4○D4 [×4], C23⋊C4 [×2], C4≀C2 [×4], C2×C42, C42⋊C2 [×3], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4, C4.9C42, C426C4, M4(2)⋊4C4, C23.C23, C2×C4≀C2, C42⋊C22, C23.37C23, C42.8D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], C23⋊Q8, C42.8D4

Character table of C42.8D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D
 size 11222811222444488888888888
ρ111111111111111111111111111    trivial
ρ211111-111111-1-1-1-1-1-11-11-1-11111    linear of order 2
ρ311111-1111111111-111-111-1-1-1-1-1    linear of order 2
ρ411111111111-1-1-1-11-1111-11-1-1-1-1    linear of order 2
ρ51111111111111111-1-1-1-1-1-1-111-1    linear of order 2
ρ611111-111111-1-1-1-1-11-11-111-111-1    linear of order 2
ρ711111-1111111111-1-1-11-1-111-1-11    linear of order 2
ρ811111111111-1-1-1-111-1-1-11-11-1-11    linear of order 2
ρ922-2-220-2-22-2222-2-200000000000    orthogonal lifted from D4
ρ1022-2-220-2-22-22-2-22200000000000    orthogonal lifted from D4
ρ11222220-2-2-2-2-200000020-2000000    orthogonal lifted from D4
ρ1222-2-22022-22-200000-2000200000    orthogonal lifted from D4
ρ1322-2-22022-22-2000002000-200000    orthogonal lifted from D4
ρ14222220-2-2-2-2-2000000-202000000    orthogonal lifted from D4
ρ1522-22-22-2-222-20000-20000000000    symplectic lifted from Q8, Schur index 2
ρ1622-22-2-2-2-222-2000020000000000    symplectic lifted from Q8, Schur index 2
ρ17222-2-20222-2-20000000000002i-2i0    complex lifted from C4○D4
ρ18222-2-20-2-2-222000000000002i00-2i    complex lifted from C4○D4
ρ19222-2-20222-2-2000000000000-2i2i0    complex lifted from C4○D4
ρ2022-22-2022-2-2200000002i00-2i0000    complex lifted from C4○D4
ρ2122-22-2022-2-220000000-2i002i0000    complex lifted from C4○D4
ρ22222-2-20-2-2-22200000000000-2i002i    complex lifted from C4○D4
ρ234-400004i-4i000-2i2i-2200000000000    complex faithful
ρ244-40000-4i4i0002i-2i-2200000000000    complex faithful
ρ254-40000-4i4i000-2i2i2-200000000000    complex faithful
ρ264-400004i-4i0002i-2i2-200000000000    complex faithful

Permutation representations of C42.8D4
On 16 points - transitive group 16T338
Generators in S16
(1 3)(2 10 6 14)(4 16 8 12)(5 7)(9 15)(11 13)
(1 11 5 15)(2 16 6 12)(3 13 7 9)(4 10 8 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 5)(2 8 6 4)(3 7)(9 13)(10 16 14 12)(11 15)

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

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

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

G:=TransitiveGroup(16,338);

On 16 points - transitive group 16T353
Generators in S16
(1 15)(2 10 6 14)(3 13)(4 16 8 12)(5 11)(7 9)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4 6 8)(10 12 14 16)

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

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

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

G:=TransitiveGroup(16,353);

Matrix representation of C42.8D4 in GL4(𝔽5) generated by

0104
3030
0400
0020
,
4030
0001
1010
0400
,
2020
0303
4030
0302
,
0030
0302
1020
0303
G:=sub<GL(4,GF(5))| [0,3,0,0,1,0,4,0,0,3,0,2,4,0,0,0],[4,0,1,0,0,0,0,4,3,0,1,0,0,1,0,0],[2,0,4,0,0,3,0,3,2,0,3,0,0,3,0,2],[0,0,1,0,0,3,0,3,3,0,2,0,0,2,0,3] >;

C42.8D4 in GAP, Magma, Sage, TeX

C_4^2._8D_4
% in TeX

G:=Group("C4^2.8D4");
// GroupNames label

G:=SmallGroup(128,763);
// by ID

G=gap.SmallGroup(128,763);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,521,718,172,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.8D4 in TeX

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