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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), D4○D8⋊2C2, C2.72(D42), C8.35(C2×D4), C4≀C2⋊5C22, D4⋊4D4⋊5C2, C8.26D4⋊4C2, C8⋊3D4⋊20C2, D4○SD16⋊1C2, D4.31(C2×D4), C8○D4⋊2C22, Q8.31(C2×D4), D4.9D4⋊5C2, D4.4D4⋊5C2, D4.3D4⋊3C2, D4.8D4⋊5C2, (C2×D8)⋊30C22, C8⋊C22⋊1C22, (C2×C4).21C24, (C2×C8).92C23, 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, C4.4D4⋊17C22, C4.10D4⋊2C22, 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

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, C4⋊1D4, 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], D4⋊4D4 [×2], D4.8D4, D4.9D4, D4.3D4 [×2], D4.4D4 [×2], C8⋊3D4, D4○D8 [×2], D4○SD16 [×2], D8⋊11D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ (1+4), D42, D8⋊11D4

Generators and relations
 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 >

Permutation representations
►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 13)(2 12)(3 11)(4 10)(5 9)(6 16)(7 15)(8 14)

(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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,16)(7,15)(8,14), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,16)(7,15)(8,14), (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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,16),(7,15),(8,14)], [(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 ►G ⊆ 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] >;
 

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

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
 

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