Copied to
clipboard

G = D8.9D4order 128 = 27

1st non-split extension by D8 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D8.9D4, C222SD32, C23.46D8, (C2×C8).63D4, C8.63(C2×D4), (C2×C4).32D8, (C2×C16)⋊7C22, (C2×SD32)⋊9C2, C2.D1610C2, C2.6(C2×SD32), C22⋊C1611C2, C2.D82C22, C4.19C22≀C2, C8.18D41C2, (C2×Q16)⋊1C22, (C22×D8).7C2, C22.97(C2×D8), C4.11(C8⋊C22), C2.8(C16⋊C22), (C2×C8).511C23, (C22×C4).346D4, C2.27(C22⋊D8), (C2×D8).108C22, (C22×C8).127C22, (C2×C4).779(C2×D4), 2-Sylow(CO-(4,7)), SmallGroup(128,919)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D8.9D4
C1C2C4C2×C4C2×C8C22×C8C22×D8 — D8.9D4
C1C2C4C2×C8 — D8.9D4
C1C22C22×C4C22×C8 — D8.9D4
C1C2C2C2C2C4C4C2×C8 — D8.9D4

Generators and relations for D8.9D4
 G = < a,b,c,d | a8=b2=1, c4=a2, d2=a4, bab=dad-1=a-1, ac=ca, cbc-1=ab, dbd-1=a5b, dcd-1=a2c3 >

Subgroups: 404 in 128 conjugacy classes, 36 normal (20 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×18], C8 [×2], C8, C2×C4 [×2], C2×C4 [×4], D4 [×10], Q8 [×2], C23, C23 [×10], C16 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×4], D8 [×6], Q16 [×2], C22×C4, C2×D4 [×9], C2×Q8, C24, Q8⋊C4, C2.D8, C2×C16 [×2], SD32 [×4], C22⋊Q8, C22×C8, C2×D8 [×2], C2×D8 [×5], C2×Q16, C22×D4, C22⋊C16, C2.D16 [×2], C8.18D4, C2×SD32 [×2], C22×D8, D8.9D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], SD32 [×2], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, C2×SD32, C16⋊C22, D8.9D4

Character table of D8.9D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111228888224161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1-1-11111-11-11111-1-1-11-111-1-11    linear of order 2
ρ41111-1-1-1-11111-1-111111-1-11-11-1-111-1    linear of order 2
ρ51111-1-111-1-111-11-11111-1-11-11-1-111-1    linear of order 2
ρ61111-1-111-1-111-1-111111-1-1-11-111-1-11    linear of order 2
ρ7111111-1-1-1-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ8111111-1-1-1-1111-1-111111111111111    linear of order 2
ρ92-2-220000-222-2000-222-20000000000    orthogonal lifted from D4
ρ102-2-2200002-22-2000-222-20000000000    orthogonal lifted from D4
ρ112-2-2200-22002-20002-2-220000000000    orthogonal lifted from D4
ρ122-2-22002-2002-20002-2-220000000000    orthogonal lifted from D4
ρ132222-2-2000022-200-2-2-2-22200000000    orthogonal lifted from D4
ρ14222222000022200-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ152222220000-2-2-20000000022-2-222-2-2    orthogonal lifted from D8
ρ162222-2-20000-2-22000000002-2-22-22-22    orthogonal lifted from D8
ρ172222-2-20000-2-2200000000-222-22-22-2    orthogonal lifted from D8
ρ182222220000-2-2-200000000-2-222-2-222    orthogonal lifted from D8
ρ1922-2-2-22000000000-2-2222-2ζ16716ζ1615169ζ165163ζ165163ζ16716ζ1615169ζ16131611ζ16131611    complex lifted from SD32
ρ2022-2-2-22000000000-2-2222-2ζ1615169ζ16716ζ16131611ζ16131611ζ1615169ζ16716ζ165163ζ165163    complex lifted from SD32
ρ2122-2-22-200000000022-2-22-2ζ16131611ζ16131611ζ16716ζ1615169ζ165163ζ165163ζ1615169ζ16716    complex lifted from SD32
ρ2222-2-22-200000000022-2-22-2ζ165163ζ165163ζ1615169ζ16716ζ16131611ζ16131611ζ16716ζ1615169    complex lifted from SD32
ρ2322-2-2-2200000000022-2-2-22ζ16131611ζ165163ζ16716ζ16716ζ16131611ζ165163ζ1615169ζ1615169    complex lifted from SD32
ρ2422-2-2-2200000000022-2-2-22ζ165163ζ16131611ζ1615169ζ1615169ζ165163ζ16131611ζ16716ζ16716    complex lifted from SD32
ρ2522-2-22-2000000000-2-222-22ζ1615169ζ1615169ζ16131611ζ165163ζ16716ζ16716ζ165163ζ16131611    complex lifted from SD32
ρ2622-2-22-2000000000-2-222-22ζ16716ζ16716ζ165163ζ16131611ζ1615169ζ1615169ζ16131611ζ165163    complex lifted from SD32
ρ274-4-44000000-4400000000000000000    orthogonal lifted from C8⋊C22
ρ284-44-40000000000022-2222-220000000000    orthogonal lifted from C16⋊C22
ρ294-44-400000000000-2222-22220000000000    orthogonal lifted from C16⋊C22

Smallest permutation representation of D8.9D4
On 32 points
Generators in S32
(1 18 5 22 9 26 13 30)(2 19 6 23 10 27 14 31)(3 20 7 24 11 28 15 32)(4 21 8 25 12 29 16 17)
(1 20)(2 4)(3 18)(5 32)(6 16)(7 30)(8 14)(9 28)(10 12)(11 26)(13 24)(15 22)(17 19)(21 31)(23 29)(25 27)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)
(1 31 9 23)(2 22 10 30)(3 29 11 21)(4 20 12 28)(5 27 13 19)(6 18 14 26)(7 25 15 17)(8 32 16 24)

G:=sub<Sym(32)| (1,18,5,22,9,26,13,30)(2,19,6,23,10,27,14,31)(3,20,7,24,11,28,15,32)(4,21,8,25,12,29,16,17), (1,20)(2,4)(3,18)(5,32)(6,16)(7,30)(8,14)(9,28)(10,12)(11,26)(13,24)(15,22)(17,19)(21,31)(23,29)(25,27), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,31,9,23)(2,22,10,30)(3,29,11,21)(4,20,12,28)(5,27,13,19)(6,18,14,26)(7,25,15,17)(8,32,16,24)>;

G:=Group( (1,18,5,22,9,26,13,30)(2,19,6,23,10,27,14,31)(3,20,7,24,11,28,15,32)(4,21,8,25,12,29,16,17), (1,20)(2,4)(3,18)(5,32)(6,16)(7,30)(8,14)(9,28)(10,12)(11,26)(13,24)(15,22)(17,19)(21,31)(23,29)(25,27), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,31,9,23)(2,22,10,30)(3,29,11,21)(4,20,12,28)(5,27,13,19)(6,18,14,26)(7,25,15,17)(8,32,16,24) );

G=PermutationGroup([(1,18,5,22,9,26,13,30),(2,19,6,23,10,27,14,31),(3,20,7,24,11,28,15,32),(4,21,8,25,12,29,16,17)], [(1,20),(2,4),(3,18),(5,32),(6,16),(7,30),(8,14),(9,28),(10,12),(11,26),(13,24),(15,22),(17,19),(21,31),(23,29),(25,27)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)], [(1,31,9,23),(2,22,10,30),(3,29,11,21),(4,20,12,28),(5,27,13,19),(6,18,14,26),(7,25,15,17),(8,32,16,24)])

Matrix representation of D8.9D4 in GL4(𝔽17) generated by

0600
14600
00160
00016
,
16200
0100
00160
0001
,
111500
1900
00016
00160
,
111500
10600
0001
0010
G:=sub<GL(4,GF(17))| [0,14,0,0,6,6,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,2,1,0,0,0,0,16,0,0,0,0,1],[11,1,0,0,15,9,0,0,0,0,0,16,0,0,16,0],[11,10,0,0,15,6,0,0,0,0,0,1,0,0,1,0] >;

D8.9D4 in GAP, Magma, Sage, TeX

D_8._9D_4
% in TeX

G:=Group("D8.9D4");
// GroupNames label

G:=SmallGroup(128,919);
// by ID

G=gap.SmallGroup(128,919);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^2,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a*b,d*b*d^-1=a^5*b,d*c*d^-1=a^2*c^3>;
// generators/relations

Export

Character table of D8.9D4 in TeX

׿
×
𝔽