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G = D8.3D4  order 128 = 27

3rd non-split extension by D8 of D4 acting via D4/C2=C22

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

Aliases: D8.3D4, D4.10D8, Q8.10D8, Q16.11D4, M4(2).18D4, M5(2)⋊4C22, D4○D8.C2, D4.C8⋊3C2, C8.68(C2×D4), C4.39(C2×D8), (C2×C16)⋊8C22, C4○D4.11D4, Q32⋊C2⋊2C2, C4.26C22≀C2, (C2×SD32)⋊12C2, M5(2)⋊C2⋊2C2, D8.C4⋊4C2, D4.5D4⋊1C2, C8○D4.3C22, C4○D8.7C22, (C2×C8).233C23, C8.C4⋊2C22, (C2×Q16)⋊11C22, (C2×D8).46C22, C2.34(C22⋊D8), C22.6(C8⋊C22), (C2×C4).41(C2×D4), SmallGroup(128,926)

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

Derived series
C1 — C2×C8 — D8.3D4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — D4○D8 — D8.3D4
Lower central
C1 — C2 — C4 — C2×C8 — D8.3D4
Upper central
C1 — C2 — C2×C4 — C8○D4 — D8.3D4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8.3D4

Generators and relations for D8.3D4
 G = < a,b,c,d | a8=b2=1, c4=a6, d2=a4, bab=dad-1=a-1, cac-1=a5, cbc-1=a5b, dbd-1=ab, dcd-1=a6c3 >

Subgroups: 304 in 108 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C16, C2×C8, C2×C8, M4(2), M4(2), D8, D8, D8, SD16, Q16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, C4.10D4, C8.C4, C2×C16, M5(2), SD32, Q32, C8○D4, C2×D8, C2×D8, C2×Q16, C4○D8, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, D4.C8, D8.C4, M5(2)⋊C2, D4.5D4, C2×SD32, Q32⋊C2, D4○D8, D8.3D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D8.3D4

Character table of D8.3D4

 class 12A2B2C2D2E2F4A4B4C4D4E8A8B8C8D8E16A16B16C16D16E16F
 size 1124888224816224816444488
ρ111111111111111111111111    trivial
ρ2111-11-1-111-11-1111-111111-1-1    linear of order 2
ρ3111-11-1-111-111111-1-1-1-1-1-111    linear of order 2
ρ411111111111-11111-1-1-1-1-1-1-1    linear of order 2
ρ5111-1-11111-1-1-1111-11-1-1-1-111    linear of order 2
ρ61111-1-1-1111-1111111-1-1-1-1-1-1    linear of order 2
ρ71111-1-1-1111-1-11111-1111111    linear of order 2
ρ8111-1-11111-1-11111-1-11111-1-1    linear of order 2
ρ922-200-22-22000-2-2200000000    orthogonal lifted from D4
ρ10222-200022-200-2-2-220000000    orthogonal lifted from D4
ρ1122-20200-220-2022-200000000    orthogonal lifted from D4
ρ1222-2002-2-22000-2-2200000000    orthogonal lifted from D4
ρ1322-20-200-2202022-200000000    orthogonal lifted from D4
ρ14222200022200-2-2-2-20000000    orthogonal lifted from D4
ρ1522-2-20002-220000000-√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ1622-220002-2-20000000-√2√2-√2√2√2-√2    orthogonal lifted from D8
ρ1722-2-20002-220000000√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ1822-220002-2-20000000√2-√2√2-√2-√2√2    orthogonal lifted from D8
ρ194440000-4-400000000000000    orthogonal lifted from C8⋊C22
ρ204-400000000002√2-2√2000ζ167+ζ165+ζ163+ζ16ζ1613+ζ1611+ζ167+ζ16ζ1615+ζ1613+ζ1611+ζ169ζ1615+ζ169+ζ165+ζ16300    complex faithful
ρ214-40000000000-2√22√2000ζ1615+ζ169+ζ165+ζ163ζ167+ζ165+ζ163+ζ16ζ1613+ζ1611+ζ167+ζ16ζ1615+ζ1613+ζ1611+ζ16900    complex faithful
ρ224-40000000000-2√22√2000ζ1613+ζ1611+ζ167+ζ16ζ1615+ζ1613+ζ1611+ζ169ζ1615+ζ169+ζ165+ζ163ζ167+ζ165+ζ163+ζ1600    complex faithful
ρ234-400000000002√2-2√2000ζ1615+ζ1613+ζ1611+ζ169ζ1615+ζ169+ζ165+ζ163ζ167+ζ165+ζ163+ζ16ζ1613+ζ1611+ζ167+ζ1600    complex faithful

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

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

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

Matrix representation of D8.3D4 ►in GL4(𝔽7) generated by

2224
1436
1062
5432
,
5446
6346
6602
2016
,
3261
3634
5401
2025
,
3532
1442
1636
5524
G:=sub<GL(4,GF(7))| [2,1,1,5,2,4,0,4,2,3,6,3,4,6,2,2],[5,6,6,2,4,3,6,0,4,4,0,1,6,6,2,6],[3,3,5,2,2,6,4,0,6,3,0,2,1,4,1,5],[3,1,1,5,5,4,6,5,3,4,3,2,2,2,6,4] >;
 

D8.3D4 in GAP, Magma, Sage, TeX 

D_8._3D_4
 
% in TeX
 
G:=Group("D8.3D4");
 
// GroupNames label
 
G:=SmallGroup(128,926);
 
// by ID
 
G=gap.SmallGroup(128,926);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,1123,570,360,2804,718,172,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^6,d^2=a^4,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^5*b,d*b*d^-1=a*b,d*c*d^-1=a^6*c^3>;
 
// generators/relations
 

Export

Character table of D8.3D4 in TeX

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