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G = D8⋊11D4  order 128 = 27

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

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

Aliases: D8⋊11D4, Q16⋊11D4, SD16⋊9D4, C42.39C23, M4(2).13C23, 2+ 1+4⋊2C22, 2- 1+4⋊2C22, C22.12+ 1+4, C2.72D42, D4○D8⋊2C2, C8.35(C2×D4), C4≀C2⋊5C22, C8.26D4⋊4C2, D4⋊4D4⋊5C2, C8⋊3D4⋊20C2, D4○SD16⋊1C2, D4.31(C2×D4), C8○D4⋊2C22, Q8.31(C2×D4), D4.8D4⋊5C2, D4.3D4⋊3C2, D4.4D4⋊5C2, D4.9D4⋊5C2, (C2×D8)⋊30C22, C8⋊C22⋊1C22, (C2×C8).92C23, (C2×C4).21C24, C4○D8.9C22, (C2×D4).7C23, C8⋊C4⋊19C22, (C2×Q8).5C23, C4⋊1D4⋊13C22, C8.C4⋊5C22, C4○D4.10C23, C4.102(C22×D4), C4.D4⋊2C22, C8.C22⋊1C22, (C2×SD16)⋊30C22, C4.10D4⋊2C22, C4.4D4⋊17C22, SmallGroup(128,2020)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — D8⋊11D4
Chief series
C1 — C2 — C4 — C2×C4 — C4○D4 — 2+ 1+4 — D4○D8 — D8⋊11D4
Lower central
C1 — C2 — C2×C4 — D8⋊11D4
Upper central
C1 — C2 — C2×C4 — D8⋊11D4
Jennings
C1 — C2 — C2 — C2×C4 — D8⋊11D4

Generators and relations for D8⋊11D4
 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, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C22⋊C4, C2×C8, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, Q16, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C8⋊C4, C4.D4, C4.D4, C4.10D4, C4≀C2, C8.C4, C4.4D4, C4⋊1D4, C8○D4, C2×D8, C2×D8, C2×SD16, C2×SD16, C4○D8, C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, 2+ 1+4, 2+ 1+4, 2- 1+4, C8.26D4, D4⋊4D4, D4.8D4, D4.9D4, D4.3D4, D4.4D4, C8⋊3D4, D4○D8, D4○SD16, D8⋊11D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, D42, D8⋊11D4

Character table of D8⋊11D4

 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 D8⋊11D4
►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 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)

(2 6)(4 8)(9 11 13 15)(10 16 14 12)

(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10), (2,6)(4,8)(9,11,13,15)(10,16,14,12), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10), (2,6)(4,8)(9,11,13,15)(10,16,14,12), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,9),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10)], [(2,6),(4,8),(9,11,13,15),(10,16,14,12)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16)]])
 
G:=TransitiveGroup(16,308);
 

Matrix representation of D8⋊11D4 ►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] >;
 

D8⋊11D4 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 D8⋊11D4 in TeX

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