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G = Q16.10D4  order 128 = 27

3rd non-split extension by Q16 of D4 acting via D4/C22=C2

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

Aliases: D8.2D4, D4.8D8, Q8.8D8, Q16.10D4, M4(2).16D4, M5(2)⋊3C22, D4○D8⋊1C2, (C2×D16)⋊3C2, D4.C8⋊1C2, C4○D4.9D4, C4.37(C2×D8), C8.66(C2×D4), C16⋊C22⋊2C2, (C2×C16)⋊2C22, C4.24C22≀C2, M5(2)⋊C2⋊1C2, D8.C4⋊1C2, D4.4D4⋊1C2, (C2×D8)⋊12C22, C8○D4.1C22, C4○D8.5C22, C8.C4⋊1C22, (C2×C8).231C23, C2.32(C22⋊D8), C22.4(C8⋊C22), (C2×C4).39(C2×D4), SmallGroup(128,924)

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

Derived series
C1 — C2×C8 — Q16.10D4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — D4○D8 — Q16.10D4
Lower central
C1 — C2 — C4 — C2×C8 — Q16.10D4
Upper central
C1 — C2 — C2×C4 — C8○D4 — Q16.10D4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — Q16.10D4

Generators and relations for Q16.10D4
 G = < a,b,c,d | a8=d2=1, b2=c4=a4, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=a3b, dcd=a4c3 >

Subgroups: 336 in 110 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, C2×D4, C4○D4, C4○D4, C4.D4, C8.C4, C2×C16, M5(2), D16, SD32, C8○D4, C2×D8, C2×D8, C4○D8, C4○D8, C8⋊C22, 2+ 1+4, D4.C8, D8.C4, M5(2)⋊C2, D4.4D4, C2×D16, C16⋊C22, D4○D8, Q16.10D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, Q16.10D4

Character table of Q16.10D4

 class 12A2B2C2D2E2F2G4A4B4C4D8A8B8C8D8E16A16B16C16D16E16F
 size 1124888162248224816444488
ρ111111111111111111111111    trivial
ρ2111-1-111111-1-1111-1-11111-1-1    linear of order 2
ρ3111-11-1-1-111-11111-111111-1-1    linear of order 2
ρ41111-1-1-1-1111-11111-1111111    linear of order 2
ρ5111-1-111-111-1-1111-11-1-1-1-111    linear of order 2
ρ61111111-111111111-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-1-11111-111111-1-1-1-1-1-1    linear of order 2
ρ8111-11-1-1111-11111-1-1-1-1-1-111    linear of order 2
ρ922-200-2202-200-2-2200000000    orthogonal lifted from D4
ρ1022-2020002-20-222-200000000    orthogonal lifted from D4
ρ1122-2002-202-200-2-2200000000    orthogonal lifted from D4
ρ1222-20-20002-20222-200000000    orthogonal lifted from D4
ρ13222-2000022-20-2-2-220000000    orthogonal lifted from D4
ρ14222200002220-2-2-2-20000000    orthogonal lifted from D4
ρ1522-220000-22-2000000√2-√2√2-√2-√2√2    orthogonal lifted from D8
ρ1622-2-20000-222000000-√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ1722-220000-22-2000000-√2√2-√2√2√2-√2    orthogonal lifted from D8
ρ1822-2-20000-222000000√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ1944400000-4-40000000000000    orthogonal lifted from C8⋊C22
ρ204-40000000000-2√22√2000-ζ167+ζ165-ζ163+ζ16ζ1615-ζ169-ζ165+ζ163ζ167-ζ165+ζ163-ζ16-ζ1615+ζ169+ζ165-ζ16300    orthogonal faithful
ρ214-40000000000-2√22√2000ζ167-ζ165+ζ163-ζ16-ζ1615+ζ169+ζ165-ζ163-ζ167+ζ165-ζ163+ζ16ζ1615-ζ169-ζ165+ζ16300    orthogonal faithful
ρ224-400000000002√2-2√2000-ζ1615+ζ169+ζ165-ζ163-ζ167+ζ165-ζ163+ζ16ζ1615-ζ169-ζ165+ζ163ζ167-ζ165+ζ163-ζ1600    orthogonal faithful
ρ234-400000000002√2-2√2000ζ1615-ζ169-ζ165+ζ163ζ167-ζ165+ζ163-ζ16-ζ1615+ζ169+ζ165-ζ163-ζ167+ζ165-ζ163+ζ1600    orthogonal faithful

Smallest permutation representation of Q16.10D4
►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 21 5 17)(2 20 6 24)(3 19 7 23)(4 18 8 22)(9 26 13 30)(10 25 14 29)(11 32 15 28)(12 31 16 27)

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

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

Matrix representation of Q16.10D4 ►in GL4(𝔽17) generated by

14300
141400
143011
140311
,
0010
116162
16000
016161
,
13042
601110
1121315
41378
,
13042
601110
61542
6400
G:=sub<GL(4,GF(17))| [14,14,14,14,3,14,3,0,0,0,0,3,0,0,11,11],[0,1,16,0,0,16,0,16,1,16,0,16,0,2,0,1],[13,6,11,4,0,0,2,13,4,11,13,7,2,10,15,8],[13,6,6,6,0,0,15,4,4,11,4,0,2,10,2,0] >;
 

Q16.10D4 in GAP, Magma, Sage, TeX 

Q_{16}._{10}D_4
 
% in TeX
 
G:=Group("Q16.10D4");
 
// GroupNames label
 
G:=SmallGroup(128,924);
 
// by ID
 
G=gap.SmallGroup(128,924);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,1123,570,360,2804,718,172,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=d^2=1,b^2=c^4=a^4,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a^3*b,d*c*d=a^4*c^3>;
 
// generators/relations
 

Export

Character table of Q16.10D4 in TeX

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