Copied to
clipboard

G = Q16.8D4order 128 = 27

1st non-split extension by Q16 of D4 acting via D4/C22=C2

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

Aliases: Q16.8D4, C222Q32, C23.47D8, (C2×Q32)⋊1C2, (C2×C4).33D8, (C2×C8).64D4, C8.64(C2×D4), C2.4(C2×Q32), C2.Q325C2, C4.20C22≀C2, C22⋊C16.5C2, (C2×C16).2C22, C22.98(C2×D8), C4.12(C8⋊C22), (C2×C8).512C23, C2.D8.3C22, C8.18D4.3C2, C2.7(Q32⋊C2), (C22×C4).347D4, (C2×Q16).1C22, (C22×Q16).7C2, C2.28(C22⋊D8), (C22×C8).128C22, (C2×C4).780(C2×D4), SmallGroup(128,920)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — Q16.8D4
C1C2C4C2×C4C2×C8C22×C8C22×Q16 — Q16.8D4
C1C2C4C2×C8 — Q16.8D4
C1C22C22×C4C22×C8 — Q16.8D4
C1C2C2C2C2C4C4C2×C8 — Q16.8D4

Generators and relations for Q16.8D4
 G = < a,b,c,d | a8=1, b2=d2=a4, c4=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=ab, dbd-1=a-1b, dcd-1=a6c3 >

Subgroups: 244 in 108 conjugacy classes, 36 normal (20 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×2], C8 [×2], C8, C2×C4 [×2], C2×C4 [×10], Q8 [×12], C23, C16 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], Q16 [×4], Q16 [×8], C22×C4, C22×C4, C2×Q8 [×10], Q8⋊C4, C2.D8, C2×C16 [×2], Q32 [×4], C22⋊Q8, C22×C8, C2×Q16, C2×Q16 [×2], C2×Q16 [×5], C22×Q8, C22⋊C16, C2.Q32 [×2], C8.18D4, C2×Q32 [×2], C22×Q16, Q16.8D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], Q32 [×2], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, C2×Q32, Q32⋊C2, Q16.8D4

Character table of Q16.8D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111222248888161622224444444444
ρ111111111111111111111111111111    trivial
ρ21111-1-111-1-111-1-111111-1-1-1-1-11-1111    linear of order 2
ρ31111-1-111-1-111-11-11111-1-1111-11-1-1-1    linear of order 2
ρ41111111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ5111111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-1-111-11-1-11-111111-1-1111-11-1-1-1    linear of order 2
ρ71111-1-111-11-1-111-11111-1-1-1-1-11-1111    linear of order 2
ρ8111111111-1-1-1-1-1-111111111111111    linear of order 2
ρ92222-2-222-2000000-2-2-2-22200000000    orthogonal lifted from D4
ρ102-2-22002-20-2002002-22-20000000000    orthogonal lifted from D4
ρ112-2-22002-2002-2000-22-220000000000    orthogonal lifted from D4
ρ122-2-22002-20200-2002-22-20000000000    orthogonal lifted from D4
ρ13222222222000000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ142-2-22002-200-22000-22-220000000000    orthogonal lifted from D4
ρ152222-2-2-2-22000000000000-2222-2-22-2    orthogonal lifted from D8
ρ16222222-2-2-20000000000002-2-222-22-2    orthogonal lifted from D8
ρ17222222-2-2-2000000000000-222-2-22-22    orthogonal lifted from D8
ρ182222-2-2-2-220000000000002-2-2-222-22    orthogonal lifted from D8
ρ192-22-2-2200000000022-2-2-22ζ165163ζ16151691615169ζ165163165163ζ16151691651631615169    symplectic lifted from Q32, Schur index 2
ρ202-22-22-2000000000-2-222-221615169ζ165163165163ζ1615169ζ16151691651631615169ζ165163    symplectic lifted from Q32, Schur index 2
ρ212-22-2-22000000000-2-2222-21615169ζ1651631651631615169ζ1615169ζ165163ζ1615169165163    symplectic lifted from Q32, Schur index 2
ρ222-22-22-200000000022-2-22-2ζ165163ζ161516916151691651631651631615169ζ165163ζ1615169    symplectic lifted from Q32, Schur index 2
ρ232-22-22-2000000000-2-222-22ζ1615169165163ζ16516316151691615169ζ165163ζ1615169165163    symplectic lifted from Q32, Schur index 2
ρ242-22-22-200000000022-2-22-21651631615169ζ1615169ζ165163ζ165163ζ16151691651631615169    symplectic lifted from Q32, Schur index 2
ρ252-22-2-22000000000-2-2222-2ζ1615169165163ζ165163ζ161516916151691651631615169ζ165163    symplectic lifted from Q32, Schur index 2
ρ262-22-2-2200000000022-2-2-221651631615169ζ1615169165163ζ1651631615169ζ165163ζ1615169    symplectic lifted from Q32, Schur index 2
ρ274-4-4400-44000000000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-40000000000022-22-22220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2944-4-400000000000-222222-220000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of Q16.8D4
On 64 points
Generators in S64
(1 17 5 21 9 25 13 29)(2 18 6 22 10 26 14 30)(3 19 7 23 11 27 15 31)(4 20 8 24 12 28 16 32)(33 59 37 63 41 51 45 55)(34 60 38 64 42 52 46 56)(35 61 39 49 43 53 47 57)(36 62 40 50 44 54 48 58)
(1 36 9 44)(2 59 10 51)(3 34 11 42)(4 57 12 49)(5 48 13 40)(6 55 14 63)(7 46 15 38)(8 53 16 61)(17 58 25 50)(18 33 26 41)(19 56 27 64)(20 47 28 39)(21 54 29 62)(22 45 30 37)(23 52 31 60)(24 43 32 35)
(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)
(1 53 9 61)(2 52 10 60)(3 51 11 59)(4 50 12 58)(5 49 13 57)(6 64 14 56)(7 63 15 55)(8 62 16 54)(17 43 25 35)(18 42 26 34)(19 41 27 33)(20 40 28 48)(21 39 29 47)(22 38 30 46)(23 37 31 45)(24 36 32 44)

G:=sub<Sym(64)| (1,17,5,21,9,25,13,29)(2,18,6,22,10,26,14,30)(3,19,7,23,11,27,15,31)(4,20,8,24,12,28,16,32)(33,59,37,63,41,51,45,55)(34,60,38,64,42,52,46,56)(35,61,39,49,43,53,47,57)(36,62,40,50,44,54,48,58), (1,36,9,44)(2,59,10,51)(3,34,11,42)(4,57,12,49)(5,48,13,40)(6,55,14,63)(7,46,15,38)(8,53,16,61)(17,58,25,50)(18,33,26,41)(19,56,27,64)(20,47,28,39)(21,54,29,62)(22,45,30,37)(23,52,31,60)(24,43,32,35), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,53,9,61)(2,52,10,60)(3,51,11,59)(4,50,12,58)(5,49,13,57)(6,64,14,56)(7,63,15,55)(8,62,16,54)(17,43,25,35)(18,42,26,34)(19,41,27,33)(20,40,28,48)(21,39,29,47)(22,38,30,46)(23,37,31,45)(24,36,32,44)>;

G:=Group( (1,17,5,21,9,25,13,29)(2,18,6,22,10,26,14,30)(3,19,7,23,11,27,15,31)(4,20,8,24,12,28,16,32)(33,59,37,63,41,51,45,55)(34,60,38,64,42,52,46,56)(35,61,39,49,43,53,47,57)(36,62,40,50,44,54,48,58), (1,36,9,44)(2,59,10,51)(3,34,11,42)(4,57,12,49)(5,48,13,40)(6,55,14,63)(7,46,15,38)(8,53,16,61)(17,58,25,50)(18,33,26,41)(19,56,27,64)(20,47,28,39)(21,54,29,62)(22,45,30,37)(23,52,31,60)(24,43,32,35), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,53,9,61)(2,52,10,60)(3,51,11,59)(4,50,12,58)(5,49,13,57)(6,64,14,56)(7,63,15,55)(8,62,16,54)(17,43,25,35)(18,42,26,34)(19,41,27,33)(20,40,28,48)(21,39,29,47)(22,38,30,46)(23,37,31,45)(24,36,32,44) );

G=PermutationGroup([(1,17,5,21,9,25,13,29),(2,18,6,22,10,26,14,30),(3,19,7,23,11,27,15,31),(4,20,8,24,12,28,16,32),(33,59,37,63,41,51,45,55),(34,60,38,64,42,52,46,56),(35,61,39,49,43,53,47,57),(36,62,40,50,44,54,48,58)], [(1,36,9,44),(2,59,10,51),(3,34,11,42),(4,57,12,49),(5,48,13,40),(6,55,14,63),(7,46,15,38),(8,53,16,61),(17,58,25,50),(18,33,26,41),(19,56,27,64),(20,47,28,39),(21,54,29,62),(22,45,30,37),(23,52,31,60),(24,43,32,35)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(1,53,9,61),(2,52,10,60),(3,51,11,59),(4,50,12,58),(5,49,13,57),(6,64,14,56),(7,63,15,55),(8,62,16,54),(17,43,25,35),(18,42,26,34),(19,41,27,33),(20,40,28,48),(21,39,29,47),(22,38,30,46),(23,37,31,45),(24,36,32,44)])

Matrix representation of Q16.8D4 in GL4(𝔽17) generated by

16000
01600
00143
001414
,
1000
01600
00125
0055
,
01600
16000
00136
001113
,
0100
1000
0017
00716
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,14,14,0,0,3,14],[1,0,0,0,0,16,0,0,0,0,12,5,0,0,5,5],[0,16,0,0,16,0,0,0,0,0,13,11,0,0,6,13],[0,1,0,0,1,0,0,0,0,0,1,7,0,0,7,16] >;

Q16.8D4 in GAP, Magma, Sage, TeX

Q_{16}._8D_4
% in TeX

G:=Group("Q16.8D4");
// GroupNames label

G:=SmallGroup(128,920);
// by ID

G=gap.SmallGroup(128,920);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,456,422,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=1,b^2=d^2=a^4,c^4=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=a^6*c^3>;
// generators/relations

Export

Character table of Q16.8D4 in TeX

׿
×
𝔽