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G = Q8.2D12order 192 = 26·3

2nd non-split extension by Q8 of D12 acting via D12/C4=S3

non-abelian, soluble

Aliases: Q8.2D12, SL2(𝔽3).2D4, (C2×C4).4S4, (C4×Q8)⋊4S3, C2.5(C4⋊S4), (C2×Q8).11D6, C22.37(C2×S4), C2.5(C4.6S4), (C4×SL2(𝔽3))⋊2C2, (C2×CSU2(𝔽3))⋊1C2, C2.5(Q8.D6), (C2×GL2(𝔽3)).2C2, (C2×SL2(𝔽3)).11C22, SmallGroup(192,954)

Series: Derived Chief Lower central Upper central

C1C2Q8C2×SL2(𝔽3) — Q8.2D12
C1C2Q8SL2(𝔽3)C2×SL2(𝔽3)C2×GL2(𝔽3) — Q8.2D12
SL2(𝔽3)C2×SL2(𝔽3) — Q8.2D12
C1C22C2×C4

Generators and relations for Q8.2D12
 G = < a,b,c,d | a4=c12=1, b2=d2=a2, bab-1=dad-1=a-1, cac-1=ab, cbc-1=a, dbd-1=a-1b, dcd-1=a2c-1 >

Subgroups: 329 in 73 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C2×C12, C22×S3, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, D6⋊C4, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), Q8.D4, C4×SL2(𝔽3), C2×CSU2(𝔽3), C2×GL2(𝔽3), Q8.2D12
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, C2×S4, C4⋊S4, Q8.D6, C4.6S4, Q8.2D12

Character table of Q8.2D12

 class 12A2B2C2D34A4B4C4D4E4F6A6B6C8A8B8C8D12A12B12C12D
 size 111124822661224888121212128888
ρ111111111111111111111111    trivial
ρ2111111-1-111-1-1111-11-11-1-1-1-1    linear of order 2
ρ31111-11-1-111-111111-11-1-1-1-1-1    linear of order 2
ρ41111-1111111-1111-1-1-1-11111    linear of order 2
ρ522220-1-2-222-20-1-1-100001111    orthogonal lifted from D6
ρ62-2-2202002-2002-2-200000000    orthogonal lifted from D4
ρ722220-1222220-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ82-2-220-1002-200-1110000-3-333    orthogonal lifted from D12
ρ92-2-220-1002-200-111000033-3-3    orthogonal lifted from D12
ρ1022-2-20-1-2i2i00001-11-2--22-2-ii-ii    complex lifted from C4.6S4
ρ1122-2-20-1-2i2i00001-112-2-2--2-ii-ii    complex lifted from C4.6S4
ρ1222-2-20-12i-2i00001-112--2-2-2i-ii-i    complex lifted from C4.6S4
ρ1322-2-20-12i-2i00001-11-2-22--2i-ii-i    complex lifted from C4.6S4
ρ143333-1033-1-1-1-100011110000    orthogonal lifted from S4
ρ1533331033-1-1-11000-1-1-1-10000    orthogonal lifted from S4
ρ163333-10-3-3-1-111000-11-110000    orthogonal lifted from C2×S4
ρ17333310-3-3-1-11-10001-11-10000    orthogonal lifted from C2×S4
ρ184-44-40-200000022-200000000    symplectic lifted from Q8.D6, Schur index 2
ρ194-44-401000000-1-110000--3-3-3--3    complex lifted from Q8.D6
ρ204-44-401000000-1-110000-3--3--3-3    complex lifted from Q8.D6
ρ2144-4-4014i-4i0000-11-10000-ii-ii    complex lifted from C4.6S4
ρ2244-4-401-4i4i0000-11-10000i-ii-i    complex lifted from C4.6S4
ρ236-6-660000-220000000000000    orthogonal lifted from C4⋊S4

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

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

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

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

Matrix representation of Q8.2D12 in GL4(𝔽73) generated by

1000
0100
004529
005628
,
1000
0100
005729
004416
,
286900
324500
004767
004753
,
286900
684500
004767
005226
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,45,56,0,0,29,28],[1,0,0,0,0,1,0,0,0,0,57,44,0,0,29,16],[28,32,0,0,69,45,0,0,0,0,47,47,0,0,67,53],[28,68,0,0,69,45,0,0,0,0,47,52,0,0,67,26] >;

Q8.2D12 in GAP, Magma, Sage, TeX

Q_8._2D_{12}
% in TeX

G:=Group("Q8.2D12");
// GroupNames label

G:=SmallGroup(192,954);
// by ID

G=gap.SmallGroup(192,954);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,708,451,1684,655,172,1013,404,285,124]);
// Polycyclic

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

Export

Character table of Q8.2D12 in TeX

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