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G = D8⋊D4order 128 = 27

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

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

Aliases: D8⋊D4, Q16⋊D4, C23.6D8, M5(2)⋊1C22, (C2×C4).6D8, C82D41C2, (C2×C8).11D4, C8.18(C2×D4), D82C41C2, C16⋊C221C2, C23.C81C2, (C2×C8).3C23, C4.Q81C22, C4.22C22≀C2, (C2×D8)⋊11C22, C4○D8.3C22, C22.19(C2×D8), D8⋊C224C2, C4.97(C8⋊C22), (C22×C4).103D4, C2.30(C22⋊D8), (C2×M4(2)).35C22, (C2×C4).278(C2×D4), SmallGroup(128,922)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D8⋊D4
C1C2C4C2×C4C2×C8C2×M4(2)D8⋊C22 — D8⋊D4
C1C2C4C2×C8 — D8⋊D4
C1C2C2×C4C2×M4(2) — D8⋊D4
C1C2C2C2C2C4C4C2×C8 — D8⋊D4

Generators and relations for D8⋊D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=dad=a-1, cac-1=a3, cbc-1=ab, dbd=a5b, dcd=c-1 >

Subgroups: 324 in 113 conjugacy classes, 32 normal (18 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×9], C8 [×2], C8, C2×C4 [×2], C2×C4 [×8], D4 [×11], Q8 [×3], C23, C23 [×2], C16 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], M4(2) [×2], D8 [×2], D8 [×3], SD16 [×4], Q16 [×2], Q16, C22×C4, C22×C4, C2×D4 [×4], C2×Q8, C4○D4 [×6], D4⋊C4, C4.Q8, M5(2) [×2], D16 [×2], SD32 [×2], C4⋊D4, C2×M4(2), C2×D8, C4○D8 [×2], C4○D8, C8⋊C22 [×2], C8.C22 [×2], C2×C4○D4, C23.C8, D82C4 [×2], C82D4, C16⋊C22 [×2], D8⋊C22, D8⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D8⋊D4

Character table of D8⋊D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F8A8B8C16A16B16C16D
 size 1124881622488164488888
ρ111111111111111111111    trivial
ρ2111-1-11111-1-11-111-1-111-1    linear of order 2
ρ3111-1-11-111-1-11111-11-1-11    linear of order 2
ρ4111111-111111-1111-1-1-1-1    linear of order 2
ρ5111-11-1-111-11-1111-1-111-1    linear of order 2
ρ61111-1-1-1111-1-1-11111111    linear of order 2
ρ71111-1-11111-1-11111-1-1-1-1    linear of order 2
ρ8111-11-1111-11-1-111-11-1-11    linear of order 2
ρ922-20200-220-2002-200000    orthogonal lifted from D4
ρ102222000222000-2-2-20000    orthogonal lifted from D4
ρ1122-20020-2200-20-2200000    orthogonal lifted from D4
ρ12222-200022-2000-2-220000    orthogonal lifted from D4
ρ1322-20-200-2202002-200000    orthogonal lifted from D4
ρ1422-200-20-220020-2200000    orthogonal lifted from D4
ρ152222000-2-2-2000000-22-22    orthogonal lifted from D8
ρ16222-2000-2-2200000022-2-2    orthogonal lifted from D8
ρ17222-2000-2-22000000-2-222    orthogonal lifted from D8
ρ182222000-2-2-20000002-22-2    orthogonal lifted from D8
ρ1944-400004-400000000000    orthogonal lifted from C8⋊C22
ρ208-8000000000000000000    orthogonal faithful

Permutation representations of D8⋊D4
On 16 points - transitive group 16T385
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)
(1 5)(2 8)(4 6)(9 10 13 14)(11 16 15 12)
(2 8)(3 7)(4 6)(9 10)(11 16)(12 15)(13 14)

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

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

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

G:=TransitiveGroup(16,385);

Matrix representation of D8⋊D4 in GL8(ℤ)

00100000
00010000
0-1000000
10000000
00000001
000000-10
00001000
00000100
,
00000001
000000-10
00001000
00000100
00100000
00010000
0-1000000
10000000
,
10000000
0-1000000
000-10000
00-100000
00000010
0000000-1
0000-1000
00000100
,
10000000
0-1000000
00010000
00100000
00000010
0000000-1
00001000
00000-100

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

D8⋊D4 in GAP, Magma, Sage, TeX

D_8\rtimes D_4
% in TeX

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

G:=SmallGroup(128,922);
// by ID

G=gap.SmallGroup(128,922);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,1123,570,521,360,1411,4037,2028,124]);
// Polycyclic

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

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Character table of D8⋊D4 in TeX

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