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G = D8.D4  order 128 = 27

1st non-split extension by D8 of D4 acting via D4/C2=C22

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

Aliases: D8.1D4, Q16.1D4, C23.7D8, M5(2)⋊2C22, (C2×C4).7D8, (C2×C8).12D4, C8.19(C2×D4), D8⋊2C4⋊2C2, C8.D4⋊1C2, Q32⋊C2⋊1C2, C23.C8⋊2C2, (C2×C8).4C23, C4.Q8⋊2C22, C4.23C22≀C2, C4○D8.4C22, C22.20(C2×D8), C4.98(C8⋊C22), (C2×Q16)⋊10C22, (C22×C4).104D4, C2.31(C22⋊D8), D8⋊C22.6C2, (C2×M4(2)).36C22, (C2×C4).279(C2×D4), SmallGroup(128,923)

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

Derived series
C1 — C2×C8 — D8.D4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C2×M4(2) — D8⋊C22 — D8.D4
Lower central
C1 — C2 — C4 — C2×C8 — D8.D4
Upper central
C1 — C2 — C2×C4 — C2×M4(2) — D8.D4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8.D4

Generators and relations for D8.D4
 G = < a,b,c,d | a8=b2=c4=1, d2=a4, bab=dad-1=a-1, cac-1=a3, cbc-1=ab, dbd-1=a5b, dcd-1=a4c-1 >

Subgroups: 276 in 108 conjugacy classes, 32 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Q8⋊C4, C4.Q8, M5(2), SD32, Q32, C22⋊Q8, C2×M4(2), C2×Q16, C4○D8, C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C23.C8, D8⋊2C4, C8.D4, Q32⋊C2, D8⋊C22, D8.D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D8.D4

Character table of D8.D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G8A8B8C16A16B16C16D
 size 1124882248816164488888
ρ111111111111111111111    trivial
ρ2111-11-111-11-1-1111-11-1-11    linear of order 2
ρ3111-1-1111-1-11-1111-1-111-1    linear of order 2
ρ41111-1-1111-1-111111-1-1-1-1    linear of order 2
ρ511111111111-1-1111-1-1-1-1    linear of order 2
ρ6111-11-111-11-11-111-1-111-1    linear of order 2
ρ71111-1-1111-1-1-1-11111111    linear of order 2
ρ8111-1-1111-1-111-111-11-1-11    linear of order 2
ρ922-200-2-2200200-2200000    orthogonal lifted from D4
ρ1022-20-20-22020002-200000    orthogonal lifted from D4
ρ1122-2020-220-20002-200000    orthogonal lifted from D4
ρ12222-20022-20000-2-220000    orthogonal lifted from D4
ρ132222002220000-2-2-20000    orthogonal lifted from D4
ρ1422-2002-2200-200-2200000    orthogonal lifted from D4
ρ15222-200-2-220000000-√2-√2√2√2    orthogonal lifted from D8
ρ16222-200-2-220000000√2√2-√2-√2    orthogonal lifted from D8
ρ17222200-2-2-20000000-√2√2-√2√2    orthogonal lifted from D8
ρ18222200-2-2-20000000√2-√2√2-√2    orthogonal lifted from D8
ρ1944-40004-4000000000000    orthogonal lifted from C8⋊C22
ρ208-8000000000000000000    symplectic faithful, Schur index 2

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

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

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

Matrix representation of D8.D4 ►in GL8(𝔽17)

00001000
00000100
1601010150
000000161
016000000
10000000
16011600160
000000160
,
00001000
000001600
1016016020
10160161611
10000000
016000000
100016010
11616116010
,
101601001610
10761000101
116910017
1016000118
10167167110
160116711016
80970017
7010160017
,
101601001610
10761000101
116910017
1016000118
160110110167
10161101671
000100017
101610017

G:=sub<GL(8,GF(17))| [0,0,16,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,16,0,0,16,16,0,0,0,1,0,0,0,0],[0,0,1,1,1,0,1,1,0,0,0,0,0,16,0,16,0,0,16,16,0,0,0,16,0,0,0,0,0,0,0,1,1,0,16,16,0,0,16,16,0,16,0,16,0,0,0,0,0,0,2,1,0,0,1,1,0,0,0,1,0,0,0,0],[10,10,1,1,1,16,8,7,16,7,16,0,0,0,0,0,0,6,9,16,16,1,9,10,1,10,1,0,7,16,7,16,0,0,0,0,16,7,0,0,0,0,0,0,7,1,0,0,16,10,1,11,1,10,1,1,10,1,7,8,10,16,7,7],[10,10,1,1,16,1,0,1,16,7,16,0,0,0,0,0,0,6,9,16,1,16,0,16,1,10,1,0,10,1,10,1,0,0,0,0,1,10,0,0,0,0,0,0,10,16,0,0,16,10,1,11,16,7,1,1,10,1,7,8,7,1,7,7] >;
 

D8.D4 in GAP, Magma, Sage, TeX 

D_8.D_4
 
% in TeX
 
G:=Group("D8.D4");
 
// GroupNames label
 
G:=SmallGroup(128,923);
 
// by ID
 
G=gap.SmallGroup(128,923);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,1123,570,521,360,1411,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^3,c*b*c^-1=a*b,d*b*d^-1=a^5*b,d*c*d^-1=a^4*c^-1>;
 
// generators/relations
 

Export

Character table of D8.D4 in TeX

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