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G = Q16.D4  order 128 = 27

2nd non-split extension by Q16 of D4 acting via D4/C2=C22

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

Aliases: D4.9D8, Q8.9D8, D8.11D4, Q16.2D4, M4(2).17D4, M5(2).1C22, Q8○D8⋊1C2, D4.C8⋊2C2, C4.38(C2×D8), C8.67(C2×D4), C16⋊C22⋊3C2, C4○D4.10D4, C4.25C22≀C2, (C2×SD32)⋊11C2, D8.C4⋊3C2, D4.4D4⋊2C2, C8.17D4⋊1C2, C4○D8.6C22, C8○D4.2C22, (C2×C16).38C22, (C2×C8).232C23, (C2×D8).45C22, C2.33(C22⋊D8), C22.5(C8⋊C22), C8.C4.3C22, (C2×Q16).44C22, (C2×C4).40(C2×D4), SmallGroup(128,925)

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

Derived series
C1 — C2×C8 — Q16.D4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — Q8○D8 — Q16.D4
Lower central
C1 — C2 — C4 — C2×C8 — Q16.D4
Upper central
C1 — C2 — C2×C4 — C8○D4 — Q16.D4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — Q16.D4

Generators and relations for Q16.D4
 G = < a,b,c,d | a8=d2=1, b2=a4, c4=a6, bab-1=dad=a-1, cac-1=a5, cbc-1=ab, dbd=a3b, dcd=a2c3 >

Subgroups: 272 in 105 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, SD16, Q16, Q16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, C4.D4, C8.C4, C2×C16, M5(2), D16, SD32, C8○D4, C2×D8, C2×Q16, C2×Q16, C4○D8, C4○D8, C8⋊C22, C8.C22, 2- 1+4, D4.C8, D8.C4, C8.17D4, D4.4D4, C2×SD32, C16⋊C22, Q8○D8, Q16.D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, Q16.D4

Character table of Q16.D4

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E16A16B16C16D16E16F
 size 1124816224888224816444488
ρ111111111111111111111111    trivial
ρ211111-11111111111-1-1-1-1-1-1-1    linear of order 2
ρ3111-11-111-1-1-11111-111111-1-1    linear of order 2
ρ4111-11111-1-1-11111-1-1-1-1-1-111    linear of order 2
ρ5111-1-1111-111-1111-1-11111-1-1    linear of order 2
ρ6111-1-1-111-111-1111-11-1-1-1-111    linear of order 2
ρ71111-1-1111-1-1-11111-1111111    linear of order 2
ρ81111-11111-1-1-111111-1-1-1-1-1-1    linear of order 2
ρ922-2000-220-220-2-2200000000    orthogonal lifted from D4
ρ1022-2020-22000-222-200000000    orthogonal lifted from D4
ρ11222-20022-2000-2-2-220000000    orthogonal lifted from D4
ρ1222-20-20-22000222-200000000    orthogonal lifted from D4
ρ13222200222000-2-2-2-20000000    orthogonal lifted from D4
ρ1422-2000-2202-20-2-2200000000    orthogonal lifted from D4
ρ1522-22002-2-200000000-√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ1622-2-2002-2200000000-√2√2-√2√2√2-√2    orthogonal lifted from D8
ρ1722-22002-2-200000000√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ1822-2-2002-2200000000√2-√2√2-√2-√2√2    orthogonal lifted from D8
ρ19444000-4-4000000000000000    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 Q16.D4
►On 32 points
Generators in S32
(1 7 13 3 9 15 5 11)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 24 30 20 26 32 22 28)

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

(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)

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

Matrix representation of Q16.D4 ►in GL4(𝔽7) generated by

2224
1436
1062
5432
,
5606
2124
2533
3155
,
0404
5222
1356
5510
,
1104
0344
0656
0665
G:=sub<GL(4,GF(7))| [2,1,1,5,2,4,0,4,2,3,6,3,4,6,2,2],[5,2,2,3,6,1,5,1,0,2,3,5,6,4,3,5],[0,5,1,5,4,2,3,5,0,2,5,1,4,2,6,0],[1,0,0,0,1,3,6,6,0,4,5,6,4,4,6,5] >;
 

Q16.D4 in GAP, Magma, Sage, TeX 

Q_{16}.D_4
 
% in TeX
 
G:=Group("Q16.D4");
 
// GroupNames label
 
G:=SmallGroup(128,925);
 
// by ID
 
G=gap.SmallGroup(128,925);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,352,1123,570,360,2804,718,172,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=d^2=1,b^2=a^4,c^4=a^6,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a*b,d*b*d=a^3*b,d*c*d=a^2*c^3>;
 
// generators/relations
 

Export

Character table of Q16.D4 in TeX

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