Copied to
clipboard

G = Q16.D4order 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○D81C2, D4.C82C2, C4.38(C2×D8), C8.67(C2×D4), C16⋊C223C2, C4○D4.10D4, C4.25C22≀C2, (C2×SD32)⋊11C2, D8.C43C2, D4.4D42C2, C8.17D41C2, 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

C1C2×C8 — Q16.D4
C1C2C4C2×C4C2×C8C8○D4Q8○D8 — Q16.D4
C1C2C4C2×C8 — Q16.D4
C1C2C2×C4C8○D4 — Q16.D4
C1C2C2C2C2C4C4C2×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 [×4], C4 [×2], C4 [×4], C22, C22 [×4], C8 [×2], C8 [×2], C2×C4, C2×C4 [×7], D4, D4 [×7], Q8, Q8 [×6], C23, C16 [×2], C2×C8, C2×C8, M4(2), M4(2) [×2], D8, D8 [×3], SD16 [×4], Q16, Q16 [×2], Q16 [×3], C2×D4, C2×Q8 [×4], C4○D4, C4○D4 [×6], C4.D4, C8.C4, C2×C16, M5(2), D16, SD32 [×3], C8○D4, C2×D8, C2×Q16, C2×Q16, C4○D8, C4○D8, C8⋊C22, C8.C22 [×3], 2- 1+4, D4.C8, D8.C4, C8.17D4, D4.4D4, C2×SD32, C16⋊C22, Q8○D8, Q16.D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], 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
ρ1522-22002-2-200000000-22-22-22    orthogonal lifted from D8
ρ1622-2-2002-2200000000-22-222-2    orthogonal lifted from D8
ρ1722-22002-2-2000000002-22-22-2    orthogonal lifted from D8
ρ1822-2-2002-22000000002-22-2-22    orthogonal lifted from D8
ρ19444000-4-4000000000000000    orthogonal lifted from C8⋊C22
ρ204-4000000000022-22000ζ16716516316ζ1613161116716ζ161516131611169ζ161516916516300    complex faithful
ρ214-40000000000-2222000ζ1615169165163ζ16716516316ζ1613161116716ζ16151613161116900    complex faithful
ρ224-40000000000-2222000ζ1613161116716ζ161516131611169ζ1615169165163ζ1671651631600    complex faithful
ρ234-4000000000022-22000ζ161516131611169ζ1615169165163ζ16716516316ζ161316111671600    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 23 29 19 25 31 21 27)(18 32 30 28 26 24 22 20)
(1 25 9 17)(2 20 10 28)(3 23 11 31)(4 18 12 26)(5 21 13 29)(6 32 14 24)(7 19 15 27)(8 30 16 22)
(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 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)

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

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

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

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

׿
×
𝔽