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G = Q8:D12order 192 = 26·3

The semidirect product of Q8 and D12 acting via D12/C4=S3

non-abelian, soluble

Aliases: Q8:D12, C4:GL2(F3), SL2(F3):1D4, (C4xQ8):2S3, C2.4(C4:S4), (C2xC4).11S4, (C2xQ8).9D6, C22.36(C2xS4), C2.4(C4.3S4), (C4xSL2(F3)):6C2, (C2xGL2(F3)):4C2, C2.4(C2xGL2(F3)), (C2xSL2(F3)).9C22, SmallGroup(192,952)

Series: Derived Chief Lower central Upper central

C1C2Q8C2xSL2(F3) — Q8:D12
C1C2Q8SL2(F3)C2xSL2(F3)C2xGL2(F3) — Q8:D12
SL2(F3)C2xSL2(F3) — Q8:D12
C1C22C2xC4

Generators and relations for Q8:D12
 G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=dad=a-1, cac-1=ab, cbc-1=a, dbd=a-1b, dcd=c-1 >

Subgroups: 473 in 91 conjugacy classes, 17 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, D4, Q8, Q8, C23, C12, D6, C2xC6, C42, C4:C4, C2xC8, SD16, C2xD4, C2xQ8, SL2(F3), D12, C2xC12, C22xS3, D4:C4, C4:C8, C4xQ8, C4:1D4, C2xSD16, GL2(F3), C2xSL2(F3), C2xD12, C4:SD16, C4xSL2(F3), C2xGL2(F3), Q8:D12
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, GL2(F3), C2xS4, C4:S4, C2xGL2(F3), C4.3S4, Q8:D12

Character table of Q8:D12

 class 12A2B2C2D2E34A4B4C4D4E6A6B6C8A8B8C8D12A12B12C12D
 size 111124248226612888121212128888
ρ111111111111111111111111    trivial
ρ21111-1-1111111111-1-1-1-11111    linear of order 2
ρ31111-111-1-111-1111-11-11-1-1-1-1    linear of order 2
ρ411111-11-1-111-11111-11-1-1-1-1-1    linear of order 2
ρ5222200-1-2-222-2-1-1-100001111    orthogonal lifted from D6
ρ622-2-200200-220-22-200000000    orthogonal lifted from D4
ρ7222200-122222-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ822-2-200-100-2201-11000033-3-3    orthogonal lifted from D12
ρ922-2-200-100-2201-110000-3-333    orthogonal lifted from D12
ρ102-22-200-12-200011-1-2-2--2--21-11-1    complex lifted from GL2(F3)
ρ112-22-200-1-2200011-1--2-2-2--2-11-11    complex lifted from GL2(F3)
ρ122-22-200-12-200011-1--2--2-2-21-11-1    complex lifted from GL2(F3)
ρ132-22-200-1-2200011-1-2--2--2-2-11-11    complex lifted from GL2(F3)
ρ143333-110-3-3-1-110001-11-10000    orthogonal lifted from C2xS4
ρ1533331-10-3-3-1-11000-11-110000    orthogonal lifted from C2xS4
ρ163333-1-1033-1-1-100011110000    orthogonal lifted from S4
ρ17333311033-1-1-1000-1-1-1-10000    orthogonal lifted from S4
ρ184-44-40014-4000-1-110000-11-11    orthogonal lifted from GL2(F3)
ρ194-4-4400-200000-22200000000    orthogonal lifted from C4.3S4
ρ204-44-4001-44000-1-1100001-11-1    orthogonal lifted from GL2(F3)
ρ214-4-44001000001-1-100003-3-33    orthogonal lifted from C4.3S4
ρ224-4-44001000001-1-10000-333-3    orthogonal lifted from C4.3S4
ρ2366-6-6000002-2000000000000    orthogonal lifted from C4:S4

Smallest permutation representation of Q8:D12
On 32 points
Generators in S32
(1 13 5 22)(2 10 6 31)(3 19 7 28)(4 16 8 25)(9 17 30 26)(11 24 32 15)(12 20 21 29)(14 27 23 18)
(1 17 5 26)(2 14 6 23)(3 11 7 32)(4 20 8 29)(9 22 30 13)(10 18 31 27)(12 25 21 16)(15 28 24 19)
(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 3)(5 7)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)

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

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

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

Matrix representation of Q8:D12 in GL4(F73) generated by

1000
0100
004121
005232
,
1000
0100
005321
004020
,
72200
72100
0001
007272
,
72000
72100
0001
0010
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,41,52,0,0,21,32],[1,0,0,0,0,1,0,0,0,0,53,40,0,0,21,20],[72,72,0,0,2,1,0,0,0,0,0,72,0,0,1,72],[72,72,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

Q8:D12 in GAP, Magma, Sage, TeX

Q_8\rtimes D_{12}
% in TeX

G:=Group("Q8:D12");
// GroupNames label

G:=SmallGroup(192,952);
// by ID

G=gap.SmallGroup(192,952);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,36,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a*b,c*b*c^-1=a,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of Q8:D12 in TeX

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