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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, (C4×Q8)⋊2S3, C2.4(C4⋊S4), (C2×C4).11S4, (C2×Q8).9D6, C22.36(C2×S4), C2.4(C4.3S4), (C4×SL2(𝔽3))⋊6C2, (C2×GL2(𝔽3))⋊4C2, C2.4(C2×GL2(𝔽3)), (C2×SL2(𝔽3)).9C22, SmallGroup(192,952)

Series: Derived Chief Lower central Upper central

C1C2Q8C2×SL2(𝔽3) — Q8⋊D12
C1C2Q8SL2(𝔽3)C2×SL2(𝔽3)C2×GL2(𝔽3) — Q8⋊D12
SL2(𝔽3)C2×SL2(𝔽3) — Q8⋊D12
C1C22C2×C4

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, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, D6, C2×C6, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, SL2(𝔽3), D12, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4×Q8, C41D4, C2×SD16, GL2(𝔽3), C2×SL2(𝔽3), C2×D12, C4⋊SD16, C4×SL2(𝔽3), C2×GL2(𝔽3), Q8⋊D12
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, GL2(𝔽3), C2×S4, C4⋊S4, C2×GL2(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 C2×S4
ρ1533331-10-3-3-1-11000-11-110000    orthogonal lifted from C2×S4
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

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Character table of Q8⋊D12 in TeX

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