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G = Q85D4order 64 = 26

1st semidirect product of Q8 and D4 acting through Inn(Q8)

p-group, metabelian, nilpotent (class 2), monomial

Aliases: Q85D4, C23.45C23, C22.42C24, C42.44C22, C2.102- 1+4, (C4×D4)⋊17C2, (C4×Q8)⋊11C2, C4.38(C2×D4), C4⋊D413C2, Q82(C22⋊C4), C22⋊Q813C2, (C22×Q8)⋊6C2, C4.4D411C2, C223(C4○D4), C4⋊C4.74C22, (C2×C4).29C23, C2.20(C22×D4), (C2×D4).68C22, C22⋊C4.6C22, (C2×Q8).62C22, (C22×C4).69C22, C22⋊C4(C2×Q8), (C2×C4○D4)⋊8C2, C2.22(C2×C4○D4), SmallGroup(64,229)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — Q85D4
C1C2C22C2×C4C22×C4C22×Q8 — Q85D4
C1C22 — Q85D4
C1C22 — Q85D4
C1C22 — Q85D4

Generators and relations for Q85D4
 G = < a,b,c,d | a4=c4=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 213 in 145 conjugacy classes, 83 normal (14 characteristic)
C1, C2 [×3], C2 [×5], C4 [×6], C4 [×8], C22, C22 [×2], C22 [×11], C2×C4 [×2], C2×C4 [×9], C2×C4 [×12], D4 [×12], Q8 [×4], Q8 [×6], C23, C23 [×3], C42 [×3], C22⋊C4, C22⋊C4 [×9], C4⋊C4 [×6], C22×C4 [×6], C2×D4 [×6], C2×Q8, C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×4], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Q85D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2- 1+4, Q85D4

Character table of Q85D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 1111224442222222222444444
ρ11111111111111111111111111    trivial
ρ21111111-11-1-1-1-111-1-1-1-1-1111-1-1    linear of order 2
ρ3111111-1-1-1-1-1111111-1-1-1-1-1111    linear of order 2
ρ4111111-11-111-1-111-1-1111-1-11-1-1    linear of order 2
ρ511111111-1-1-11-1-1-11-1-1-111-1-11-1    linear of order 2
ρ61111111-1-111-11-1-1-1111-11-1-1-11    linear of order 2
ρ7111111-1-11111-1-1-11-111-1-11-11-1    linear of order 2
ρ8111111-111-1-1-11-1-1-11-1-11-11-1-11    linear of order 2
ρ91111-1-1-11-11-1-1-111-1-11-1-111-111    linear of order 2
ρ101111-1-1-1-1-1-11111111-11111-1-1-1    linear of order 2
ρ111111-1-11-11-11-1-111-1-1-111-1-1-111    linear of order 2
ρ121111-1-11111-11111111-1-1-1-1-1-1-1    linear of order 2
ρ131111-1-1-111-11-11-1-1-11-11-11-111-1    linear of order 2
ρ141111-1-1-1-111-11-1-1-11-11-111-11-11    linear of order 2
ρ151111-1-11-1-11-1-11-1-1-111-11-1111-1    linear of order 2
ρ161111-1-111-1-111-1-1-11-1-11-1-111-11    linear of order 2
ρ172-2-22000000022-22-2-200000000    orthogonal lifted from D4
ρ182-2-220000000-222-22-200000000    orthogonal lifted from D4
ρ192-2-2200000002-22-2-2200000000    orthogonal lifted from D4
ρ202-2-220000000-2-2-222200000000    orthogonal lifted from D4
ρ212-22-2-220002i-2i000000-2i2i000000    complex lifted from C4○D4
ρ222-22-22-20002i2i000000-2i-2i000000    complex lifted from C4○D4
ρ232-22-2-22000-2i2i0000002i-2i000000    complex lifted from C4○D4
ρ242-22-22-2000-2i-2i0000002i2i000000    complex lifted from C4○D4
ρ2544-4-4000000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of Q85D4
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 24 3 22)(2 23 4 21)(5 14 7 16)(6 13 8 15)(9 27 11 25)(10 26 12 28)(17 31 19 29)(18 30 20 32)
(1 5 12 30)(2 6 9 31)(3 7 10 32)(4 8 11 29)(13 25 19 21)(14 26 20 22)(15 27 17 23)(16 28 18 24)
(1 26)(2 27)(3 28)(4 25)(5 14)(6 15)(7 16)(8 13)(9 23)(10 24)(11 21)(12 22)(17 31)(18 32)(19 29)(20 30)

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

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

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

Q85D4 is a maximal subgroup of
SD16⋊D4  SD168D4  Q169D4  Q1610D4  SD162D4  Q164D4  SD1610D4  Q1612D4  C42.465C23  C42.466C23  C42.43C23  C42.48C23  C42.51C23  C42.56C23  C42.475C23  C42.476C23  D4×C4○D4  C22.69C25  C22.75C25  C22.76C25  C22.77C25  C22.78C25  C4⋊2- 1+4  C22.89C25  C22.94C25  C22.103C25  C22.105C25  C23.144C24  C22.113C25  C22.125C25  C22.129C25  C22.130C25  C22.140C25  C22.147C25  C22.150C25  C22.151C25  SL2(𝔽3)⋊5D4  Q84S4
 C2p.2- 1+4: C22.71C25  C22.100C25  C22.107C25  C22.111C25  C22.136C25  C22.141C25  C22.143C25  Dic623D4 ...
Q85D4 is a maximal quotient of
Q8×C22⋊C4  C23.223C24  C23.233C24  C23.234C24  C24.215C23  C23.244C24  C24.220C23  C24.244C23  C23.309C24  C23.315C24  C23.316C24  C23.321C24  C24.259C23  C23.327C24  C23.329C24  C24.264C23  C24.565C23  C24.267C23  C24.569C23  C24.269C23  C23.346C24  C23.348C24  C23.352C24  C23.353C24  C24.279C23  C23.360C24  C24.282C23  C24.283C23  C23.369C24  C23.375C24  C23.377C24  C24.573C23  C24.301C23  C23.391C24  C23.419C24  C42.165D4  C4219D4  C42.168D4  C426Q8  C23.458C24  C24.332C23  C23.574C24  C23.576C24  C23.581C24  C23.590C24  C24.403C23  C23.602C24  C23.607C24  C23.611C24  C23.616C24  C23.619C24  C24.418C23  C23.625C24  C24.420C23  C24.421C23  C23.630C24  C23.631C24
 Q8⋊D4p: Q84D8  Q86D12  Q85D20  Q85D28 ...
 C2p.2- 1+4: Q87SD16  C42.501C23  C42.502C23  Q88SD16  Q85Q16  C42.505C23  C42.506C23  C42.507C23 ...

Matrix representation of Q85D4 in GL4(𝔽5) generated by

1000
0100
0001
0040
,
1000
0100
0003
0030
,
3200
0200
0003
0020
,
4000
3100
0003
0020
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0],[3,0,0,0,2,2,0,0,0,0,0,2,0,0,3,0],[4,3,0,0,0,1,0,0,0,0,0,2,0,0,3,0] >;

Q85D4 in GAP, Magma, Sage, TeX

Q_8\rtimes_5D_4
% in TeX

G:=Group("Q8:5D4");
// GroupNames label

G:=SmallGroup(64,229);
// by ID

G=gap.SmallGroup(64,229);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650,297,69]);
// Polycyclic

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

Export

Character table of Q85D4 in TeX

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