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G = D4.D8  order 128 = 27

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

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

Aliases: D4.6D8, D4⋊3SD16, C42.200C23, D42.2C2, D4⋊C8⋊20C2, (C4×C8)⋊3C22, C4⋊C4.54D4, C4.28(C2×D8), C4⋊Q8⋊2C22, C4⋊C8⋊43C22, D4⋊Q8⋊1C2, C4.4D8⋊4C2, C4.D8⋊8C2, (C2×D4).254D4, C4.31(C2×SD16), D4.D4⋊30C2, C4.61(C8⋊C22), (C4×D4).31C22, C2.14(C22⋊D8), C4⋊1D4.17C22, C2.14(C22⋊SD16), C22.166C22≀C2, C2.12(D4.9D4), (C2×C4).957(C2×D4), SmallGroup(128,371)

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=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 464 in 154 conjugacy classes, 38 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C24, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C22≀C2, C4⋊D4, C4⋊1D4, C4⋊Q8, C2×SD16, C22×D4, D4⋊C8, C4.D8, D4.D4, D4⋊Q8, C4.4D8, D42, D4.D8
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C22≀C2, C2×D8, C2×SD16, C8⋊C22, C22⋊D8, C22⋊SD16, D4.9D4, D4.D8

Character table of D4.D8

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 111144448822224881644448888
ρ111111111111111111111111111    trivial
ρ211111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-11-1-1-1111111-11-1-1-1-1-111-1    linear of order 2
ρ411111-11-1-1-1111111-1-111111-1-11    linear of order 2
ρ51111-1-1-1-11111111-1-1-1-1-1-1-11111    linear of order 2
ρ61111-1-1-1-11111111-1-111111-1-1-1-1    linear of order 2
ρ71111-11-11-1-111111-11-11111-111-1    linear of order 2
ρ81111-11-11-1-111111-111-1-1-1-11-1-11    linear of order 2
ρ922220000-22-2-2-2-2200000000000    orthogonal lifted from D4
ρ10222220200022-2-2-2-20000000000    orthogonal lifted from D4
ρ1122220-20-200-2-222-202000000000    orthogonal lifted from D4
ρ122222-20-200022-2-2-220000000000    orthogonal lifted from D4
ρ132222020200-2-222-20-2000000000    orthogonal lifted from D4
ρ14222200002-2-2-2-2-2200000000000    orthogonal lifted from D4
ρ1522-2-2020-20000-220000√2-√2√2-√20-√2√20    orthogonal lifted from D8
ρ1622-2-20-2020000-220000-√2√2-√2√20-√2√20    orthogonal lifted from D8
ρ1722-2-2020-20000-220000-√2√2-√2√20√2-√20    orthogonal lifted from D8
ρ1822-2-20-2020000-220000√2-√2√2-√20√2-√20    orthogonal lifted from D8
ρ192-2-2220-20002-2000000√-2-√-2-√-2√-2-√-200√-2    complex lifted from SD16
ρ202-2-22-2020002-2000000√-2-√-2-√-2√-2√-200-√-2    complex lifted from SD16
ρ212-2-2220-20002-2000000-√-2√-2√-2-√-2√-200-√-2    complex lifted from SD16
ρ222-2-22-2020002-2000000-√-2√-2√-2-√-2-√-200√-2    complex lifted from SD16
ρ2344-4-4000000004-4000000000000    orthogonal lifted from C8⋊C22
ρ244-4-44000000-4400000000000000    orthogonal lifted from C8⋊C22
ρ254-44-4000000000000002i2i-2i-2i0000    complex lifted from D4.9D4
ρ264-44-400000000000000-2i-2i2i2i0000    complex lifted from D4.9D4

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

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

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

0100
16000
0010
0001
,
0100
1000
00160
00016
,
51200
5500
0033
00143
,
51200
121200
0033
00314
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[5,5,0,0,12,5,0,0,0,0,3,14,0,0,3,3],[5,12,0,0,12,12,0,0,0,0,3,3,0,0,3,14] >;
 

D4.D8 in GAP, Magma, Sage, TeX 

D_4.D_8
 
% in TeX
 
G:=Group("D4.D8");
 
// GroupNames label
 
G:=SmallGroup(128,371);
 
// by ID
 
G=gap.SmallGroup(128,371);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,1123,570,521,136,2804,1411,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
 
// generators/relations
 

Export

Character table of D4.D8 in TeX

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