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G = Q86D4order 64 = 26

2nd semidirect product of Q8 and D4 acting through Inn(Q8)

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

Aliases: Q86D4, C23.17C23, C42.46C22, C22.44C24, C2.142+ 1+4, Q82(C4⋊C4), (C4×D4)⋊18C2, C43(C4○D4), C41D48C2, (C4×Q8)⋊13C2, C4.40(C2×D4), C4⋊D414C2, C4⋊C4.82C22, (C2×C4).55C23, C2.22(C22×D4), (C2×D4).69C22, (C2×Q8).74C22, C22⋊C4.21C22, (C22×C4).71C22, C4⋊C4(C2×Q8), (C2×C4○D4)⋊9C2, C2.23(C2×C4○D4), SmallGroup(64,231)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — Q86D4
C1C2C22C2×C4C22×C4C2×C4○D4 — Q86D4
C1C22 — Q86D4
C1C22 — Q86D4
C1C22 — Q86D4

Generators and relations for Q86D4
 G = < a,b,c,d | a4=c4=d2=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 253 in 156 conjugacy classes, 83 normal (10 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×5], C22, C22 [×18], C2×C4, C2×C4 [×8], C2×C4 [×12], D4 [×24], Q8 [×4], C23 [×6], C42 [×3], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4 [×6], C2×D4 [×15], C2×Q8, C4○D4 [×8], C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C41D4 [×3], C2×C4○D4 [×2], Q86D4
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, Q86D4

Character table of Q86D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O
 size 1111444444222222222222444
ρ11111111111111111111111111    trivial
ρ211111-1-1-11-11111-1-11-11-1111-1-1    linear of order 2
ρ31111-1-11-111-1-111-1-1-1-11-11-1-111    linear of order 2
ρ41111-11-111-1-1-11111-11111-1-1-1-1    linear of order 2
ρ511111-1-11-11-1-1-11-11-1-111-1-11-11    linear of order 2
ρ61111111-1-1-1-1-1-111-1-111-1-1-111-1    linear of order 2
ρ71111-11-1-1-1111-111-1111-1-11-1-11    linear of order 2
ρ81111-1-111-1-111-11-111-111-11-11-1    linear of order 2
ρ91111-1-1-1111-1-1-1-11-111-1-1-1111-1    linear of order 2
ρ101111-111-11-1-1-1-1-1-111-1-11-111-11    linear of order 2
ρ11111111-1-11111-1-1-11-1-1-11-1-1-11-1    linear of order 2
ρ1211111-1111-111-1-11-1-11-1-1-1-1-1-11    linear of order 2
ρ131111-1111-11111-1-1-1-1-1-1-11-11-1-1    linear of order 2
ρ141111-1-1-1-1-1-1111-111-11-111-1111    linear of order 2
ρ1511111-11-1-11-1-11-11111-1111-1-1-1    linear of order 2
ρ16111111-11-1-1-1-11-1-1-11-1-1-111-111    linear of order 2
ρ172-2-2200000000-202-20-20220000    orthogonal lifted from D4
ρ182-2-220000000020220-20-2-20000    orthogonal lifted from D4
ρ192-2-2200000000-20-22020-220000    orthogonal lifted from D4
ρ202-2-220000000020-2-20202-20000    orthogonal lifted from D4
ρ212-22-2000000-2202i00-2i0-2i002i000    complex lifted from C4○D4
ρ222-22-20000002-202i002i0-2i00-2i000    complex lifted from C4○D4
ρ232-22-20000002-20-2i00-2i02i002i000    complex lifted from C4○D4
ρ242-22-2000000-220-2i002i02i00-2i000    complex lifted from C4○D4
ρ2544-4-4000000000000000000000    orthogonal lifted from 2+ 1+4

Smallest permutation representation of Q86D4
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 16 3 14)(2 15 4 13)(5 24 7 22)(6 23 8 21)(9 19 11 17)(10 18 12 20)(25 29 27 31)(26 32 28 30)
(1 31 12 8)(2 30 9 7)(3 29 10 6)(4 32 11 5)(13 26 17 22)(14 25 18 21)(15 28 19 24)(16 27 20 23)
(1 28)(2 25)(3 26)(4 27)(5 20)(6 17)(7 18)(8 19)(9 21)(10 22)(11 23)(12 24)(13 29)(14 30)(15 31)(16 32)

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

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

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

Q86D4 is a maximal subgroup of
Q82SD16  Q8.D8  Q83SD16  Q88SD16  C42.507C23  C42.509C23  D4×C4○D4  C22.69C25  C22.76C25  C22.77C25  C4⋊2- 1+4  C22.87C25  C22.111C25  C22.113C25  SL2(𝔽3)⋊6D4
 Q8⋊D4p: Q8⋊D8  Q84D8  Q85D8  Q87D12  Q86D20  Q86D28 ...
 C2p.2+ 1+4: SD167D4  Q1610D4  SD161D4  Q165D4  SD1611D4  Q1613D4  C42.469C23  C42.470C23 ...
Q86D4 is a maximal quotient of
C23.223C24  Q8×C4⋊C4  C23.236C24  C24.217C23  C24.219C23  C24.244C23  C24.249C23  C23.323C24  C24.259C23  C23.328C24  C24.263C23  C24.268C23  C23.345C24  C23.348C24  C23.354C24  C23.356C24  C23.367C24  C24.290C23  C23.379C24  C23.390C24  C23.412C24  C42.166D4  C4220D4  C42.171D4  C42.35Q8  C24.327C23  C23.573C24  C24.389C23  C24.401C23  C23.605C24  C24.413C23  C23.621C24  C23.627C24  C23.630C24  C23.632C24  C23.633C24
 Q8⋊D4p: Q85D8  Q87D12  Q86D20  Q86D28 ...
 C2p.2+ 1+4: Q89SD16  C42.527C23  C42.528C23  Q86Q16  C42.530C23  C42.72C23  C42.73C23  C42.74C23 ...

Matrix representation of Q86D4 in GL4(𝔽5) generated by

3000
0200
0040
0004
,
0100
4000
0040
0004
,
0200
2000
0012
0044
,
4000
0100
0043
0001
G:=sub<GL(4,GF(5))| [3,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4],[0,4,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[0,2,0,0,2,0,0,0,0,0,1,4,0,0,2,4],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,3,1] >;

Q86D4 in GAP, Magma, Sage, TeX

Q_8\rtimes_6D_4
% in TeX

G:=Group("Q8:6D4");
// GroupNames label

G:=SmallGroup(64,231);
// by ID

G=gap.SmallGroup(64,231);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650,86,297,69]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^4=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=a^-1,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 Q86D4 in TeX

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