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G = D811D4order 128 = 27

5th semidirect product of D8 and D4 acting via D4/C22=C2

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

Aliases: D811D4, Q1611D4, SD169D4, C42.39C23, M4(2).13C23, 2+ 1+42C22, 2- 1+42C22, C22.12+ 1+4, C2.72D42, D4○D82C2, C8.35(C2×D4), C4≀C25C22, C8.26D44C2, D44D45C2, C83D420C2, D4○SD161C2, D4.31(C2×D4), C8○D42C22, Q8.31(C2×D4), D4.8D45C2, D4.3D43C2, D4.4D45C2, D4.9D45C2, (C2×D8)⋊30C22, C8⋊C221C22, (C2×C8).92C23, (C2×C4).21C24, C4○D8.9C22, (C2×D4).7C23, C8⋊C419C22, (C2×Q8).5C23, C41D413C22, C8.C45C22, C4○D4.10C23, C4.102(C22×D4), C4.D42C22, C8.C221C22, (C2×SD16)⋊30C22, C4.10D42C22, C4.4D417C22, SmallGroup(128,2020)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D811D4
C1C2C4C2×C4C4○D42+ 1+4D4○D8 — D811D4
C1C2C2×C4 — D811D4
C1C2C2×C4 — D811D4
C1C2C2C2×C4 — D811D4

Generators and relations for D811D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, cac-1=a5, dad=a3, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 524 in 232 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2 [×8], C4 [×2], C4 [×6], C22, C22 [×13], C8 [×4], C8 [×4], C2×C4, C2×C4 [×11], D4 [×4], D4 [×19], Q8 [×4], Q8 [×3], C23 [×6], C42, C22⋊C4, C2×C8 [×2], C2×C8 [×4], M4(2) [×4], M4(2) [×4], D8 [×2], D8 [×10], SD16 [×4], SD16 [×12], Q16 [×2], Q16 [×2], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C2×Q8 [×2], C4○D4 [×4], C4○D4 [×9], C8⋊C4, C4.D4, C4.D4 [×2], C4.10D4, C4≀C2 [×4], C8.C4 [×2], C4.4D4, C41D4, C8○D4 [×4], C2×D8 [×2], C2×D8 [×2], C2×SD16 [×2], C2×SD16 [×2], C4○D8 [×2], C4○D8 [×4], C8⋊C22 [×6], C8⋊C22 [×6], C8.C22 [×2], C8.C22 [×2], 2+ 1+4, 2+ 1+4 [×2], 2- 1+4, C8.26D4 [×2], D44D4 [×2], D4.8D4, D4.9D4, D4.3D4 [×2], D4.4D4 [×2], C83D4, D4○D8 [×2], D4○SD16 [×2], D811D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ 1+4, D42, D811D4

Character table of D811D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 11244448882244448844448888
ρ111111111111111111111111111    trivial
ρ2111-11-1-11-11111-1-1-1-1-1-111-1-1111    linear of order 2
ρ3111-1-1-1-111111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ41111-1111-1111-1111-1-1-111-11-1-1-1    linear of order 2
ρ5111-1-111-1-1-111-11-111-11111-111-1    linear of order 2
ρ61111-1-1-1-11-111-1-11-1-11-111-1111-1    linear of order 2
ρ711111-1-1-1-1-1111-11-11-111111-1-11    linear of order 2
ρ8111-1111-11-11111-11-11-111-1-1-1-11    linear of order 2
ρ9111-111-1-1-11111-1-1111-1-1-1-111-1-1    linear of order 2
ρ1011111-11-11111111-1-1-11-1-11-11-1-1    linear of order 2
ρ111111-1-11-1-1111-111-111-1-1-1-1-1-111    linear of order 2
ρ12111-1-11-1-11111-1-1-11-1-11-1-111-111    linear of order 2
ρ131111-11-111-111-1-1111-1-1-1-1-1-11-11    linear of order 2
ρ14111-1-1-111-1-111-11-1-1-111-1-1111-11    linear of order 2
ρ15111-11-1111-11111-1-11-1-1-1-1-11-11-1    linear of order 2
ρ16111111-11-1-1111-111-111-1-11-1-11-1    linear of order 2
ρ1722-20022000-220-20-2000-2200000    orthogonal lifted from D4
ρ1822-200-22000-220-2020002-200000    orthogonal lifted from D4
ρ1922-2-22000002-2-202000-20020000    orthogonal lifted from D4
ρ2022-22-2000002-220-2000-20020000    orthogonal lifted from D4
ρ2122-2-2-2000002-2202000200-20000    orthogonal lifted from D4
ρ2222-222000002-2-20-2000200-20000    orthogonal lifted from D4
ρ2322-2002-2000-22020-20002-200000    orthogonal lifted from D4
ρ2422-200-2-2000-220202000-2200000    orthogonal lifted from D4
ρ254440000000-4-400000000000000    orthogonal lifted from 2+ 1+4
ρ268-8000000000000000000000000    orthogonal faithful

Permutation representations of D811D4
On 16 points - transitive group 16T308
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)
(2 6)(4 8)(9 11 13 15)(10 16 14 12)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)

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

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

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

G:=TransitiveGroup(16,308);

Matrix representation of D811D4 in GL8(ℤ)

0000000-1
000000-10
00001000
00000-100
000-10000
00-100000
10000000
0-1000000
,
00001000
00000-100
0000000-1
000000-10
10000000
0-1000000
000-10000
00-100000
,
-10000000
0-1000000
000-10000
00100000
00000-100
00001000
00000010
00000001
,
-10000000
01000000
00-100000
00010000
00000-100
0000-1000
0000000-1
000000-10

G:=sub<GL(8,Integers())| [0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0] >;

D811D4 in GAP, Magma, Sage, TeX

D_8\rtimes_{11}D_4
% in TeX

G:=Group("D8:11D4");
// GroupNames label

G:=SmallGroup(128,2020);
// by ID

G=gap.SmallGroup(128,2020);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,346,2804,1411,375,172,4037,1027,124]);
// Polycyclic

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

Export

Character table of D811D4 in TeX

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