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G = D4⋊D8  order 128 = 27

1st semidirect product of D4 and D8 acting via D8/D4=C2

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

Aliases: D4⋊2D8, C42.180C23, D42⋊1C2, D4⋊C8⋊7C2, C4⋊D8⋊1C2, C8⋊4D4⋊1C2, C4⋊C8⋊1C22, (C4×C8)⋊6C22, C4⋊C4.46D4, C4.24(C2×D8), C4.D8⋊2C2, (C2×D4).248D4, C4⋊1D4⋊1C22, C4.54(C8⋊C22), (C4×D4).17C22, C2.10(C22⋊D8), C2.12(D4⋊4D4), C22.146C22≀C2, (C2×C4).937(C2×D4), 2-Sylow(PSO+(4,7)), SmallGroup(128,351)

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

Derived series
C1 — C42 — D4⋊D8
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — D42 — D4⋊D8
Lower central
C1 — C22 — C42 — D4⋊D8
Upper central
C1 — C22 — C42 — D4⋊D8
Jennings
C1 — C22 — C22 — C42 — D4⋊D8

Generators and relations for D4⋊D8
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=dbd=ab, dcd=c-1 >

Subgroups: 536 in 168 conjugacy classes, 38 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C4×C8, D4⋊C4, C4⋊C8, C4×D4, C22≀C2, C4⋊D4, C4⋊1D4, C2×D8, C22×D4, D4⋊C8, C4.D8, C4⋊D8, C8⋊4D4, D42, D4⋊D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D4⋊4D4, D4⋊D8

Character table of D4⋊D8

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H
 size 111144448816222248844448888
ρ111111111111111111111111111    trivial
ρ211111-1-11-1-1-111111-111111-1-111    linear of order 2
ρ31111-111-1-1-1-1111111-1111111-1-1    linear of order 2
ρ41111-1-1-1-111111111-1-11111-1-1-1-1    linear of order 2
ρ51111-1-1-1-111-111111-1-1-1-1-1-11111    linear of order 2
ρ61111-111-1-1-11111111-1-1-1-1-1-1-111    linear of order 2
ρ711111-1-11-1-1111111-11-1-1-1-111-1-1    linear of order 2
ρ81111111111-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ92222-200-200022-2-2-20200000000    orthogonal lifted from D4
ρ10222200002-20-2-2-2-220000000000    orthogonal lifted from D4
ρ1122220220000-2-222-2-2000000000    orthogonal lifted from D4
ρ1222220-2-20000-2-222-22000000000    orthogonal lifted from D4
ρ132222200200022-2-2-20-200000000    orthogonal lifted from D4
ρ1422220000-220-2-2-2-220000000000    orthogonal lifted from D4
ρ1522-2-20-22000000-22000√2-√2-√2√2-√2√200    orthogonal lifted from D8
ρ162-2-22-20020002-200000-√2-√2√2√200-√2√2    orthogonal lifted from D8
ρ1722-2-20-22000000-22000-√2√2√2-√2√2-√200    orthogonal lifted from D8
ρ1822-2-202-2000000-22000√2-√2-√2√2√2-√200    orthogonal lifted from D8
ρ192-2-22-20020002-200000√2√2-√2-√200√2-√2    orthogonal lifted from D8
ρ2022-2-202-2000000-22000-√2√2√2-√2-√2√200    orthogonal lifted from D8
ρ212-2-22200-20002-200000√2√2-√2-√200-√2√2    orthogonal lifted from D8
ρ222-2-22200-20002-200000-√2-√2√2√200√2-√2    orthogonal lifted from D8
ρ234-44-400000000000000-22-220000    orthogonal lifted from D4⋊4D4
ρ2444-4-40000000004-400000000000    orthogonal lifted from C8⋊C22
ρ254-4-440000000-440000000000000    orthogonal lifted from C8⋊C22
ρ264-44-4000000000000002-22-20000    orthogonal lifted from D4⋊4D4

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

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

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

Matrix representation of D4⋊D8 ►in GL4(𝔽17) generated by

1000
0100
001615
0011
,
16000
01600
0010
001616
,
141400
31400
0006
0030
,
31400
141400
00011
00140
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,16,1,0,0,15,1],[16,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16],[14,3,0,0,14,14,0,0,0,0,0,3,0,0,6,0],[3,14,0,0,14,14,0,0,0,0,0,14,0,0,11,0] >;
 

D4⋊D8 in GAP, Magma, Sage, TeX 

D_4\rtimes D_8
 
% in TeX
 
G:=Group("D4:D8");
 
// GroupNames label
 
G:=SmallGroup(128,351);
 
// by ID
 
G=gap.SmallGroup(128,351);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,422,1123,570,521,136,2804,1411,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=a*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of D4⋊D8 in TeX

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