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G = Q8×A4order 96 = 25·3

Direct product of Q8 and A4

direct product, metabelian, soluble, monomial

Aliases: Q8×A4, C22⋊(C3×Q8), C4.1(C2×A4), (C22×C4).C6, (C4×A4).3C2, (C22×Q8)⋊2C3, C23.7(C2×C6), C2.4(C22×A4), (C2×A4).8C22, SmallGroup(96,199)

Series: Derived Chief Lower central Upper central

C1C23 — Q8×A4
C1C22C23C2×A4C4×A4 — Q8×A4
C22C23 — Q8×A4
C1C2Q8

Generators and relations for Q8×A4
 G = < a,b,c,d,e | a4=c2=d2=e3=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

3C2
3C2
4C3
3C22
3C4
3C22
3C4
3C4
4C6
3C2×C4
3C2×C4
3Q8
3Q8
3Q8
3C2×C4
3C2×C4
3C2×C4
3C2×C4
3Q8
3Q8
4C12
4C12
4C12
3C2×Q8
3C2×Q8
3C2×Q8
3C2×Q8
4C3×Q8

Character table of Q8×A4

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B12A12B12C12D12E12F
 size 11334422266644888888
ρ111111111111111111111    trivial
ρ21111111-1-1-11-111-1-111-1-1    linear of order 2
ρ3111111-1-111-1-1111-1-1-11-1    linear of order 2
ρ4111111-11-1-1-1111-11-1-1-11    linear of order 2
ρ51111ζ3ζ32-11-1-1-11ζ3ζ32ζ6ζ32ζ65ζ6ζ65ζ3    linear of order 6
ρ61111ζ3ζ32111111ζ3ζ32ζ32ζ32ζ3ζ32ζ3ζ3    linear of order 3
ρ71111ζ32ζ31-1-1-11-1ζ32ζ3ζ65ζ65ζ32ζ3ζ6ζ6    linear of order 6
ρ81111ζ32ζ3111111ζ32ζ3ζ3ζ3ζ32ζ3ζ32ζ32    linear of order 3
ρ91111ζ3ζ321-1-1-11-1ζ3ζ32ζ6ζ6ζ3ζ32ζ65ζ65    linear of order 6
ρ101111ζ32ζ3-1-111-1-1ζ32ζ3ζ3ζ65ζ6ζ65ζ32ζ6    linear of order 6
ρ111111ζ3ζ32-1-111-1-1ζ3ζ32ζ32ζ6ζ65ζ6ζ3ζ65    linear of order 6
ρ121111ζ32ζ3-11-1-1-11ζ32ζ3ζ65ζ3ζ6ζ65ζ6ζ32    linear of order 6
ρ132-22-222000000-2-2000000    symplectic lifted from Q8, Schur index 2
ρ142-22-2-1+-3-1--30000001--31+-3000000    complex lifted from C3×Q8
ρ152-22-2-1--3-1+-30000001+-31--3000000    complex lifted from C3×Q8
ρ1633-1-100-33-311-100000000    orthogonal lifted from C2×A4
ρ1733-1-100333-1-1-100000000    orthogonal lifted from A4
ρ1833-1-100-3-33-11100000000    orthogonal lifted from C2×A4
ρ1933-1-1003-3-31-1100000000    orthogonal lifted from C2×A4
ρ206-6-220000000000000000    symplectic faithful, Schur index 2

Permutation representations of Q8×A4
On 24 points - transitive group 24T86
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 23 3 21)(2 22 4 24)(5 16 7 14)(6 15 8 13)(9 19 11 17)(10 18 12 20)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(1 16 11)(2 13 12)(3 14 9)(4 15 10)(5 19 21)(6 20 22)(7 17 23)(8 18 24)

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

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), (1,23,3,21)(2,22,4,24)(5,16,7,14)(6,15,8,13)(9,19,11,17)(10,18,12,20), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (1,16,11)(2,13,12)(3,14,9)(4,15,10)(5,19,21)(6,20,22)(7,17,23)(8,18,24) );

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)], [(1,23,3,21),(2,22,4,24),(5,16,7,14),(6,15,8,13),(9,19,11,17),(10,18,12,20)], [(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)], [(1,3),(2,4),(9,11),(10,12),(17,19),(18,20),(21,23),(22,24)], [(1,16,11),(2,13,12),(3,14,9),(4,15,10),(5,19,21),(6,20,22),(7,17,23),(8,18,24)])

G:=TransitiveGroup(24,86);

Q8×A4 is a maximal subgroup of   A42Q16  Q83S4  Q84S4
Q8×A4 is a maximal quotient of   SL2(𝔽3)⋊3Q8

Matrix representation of Q8×A4 in GL5(𝔽13)

1212000
21000
001200
000120
000012
,
80000
105000
001200
000120
000012
,
10000
01000
001200
001201
001210
,
10000
01000
000121
000120
001120
,
90000
09000
00009
00900
00090

G:=sub<GL(5,GF(13))| [12,2,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,10,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[9,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,9,0,0,9,0,0] >;

Q8×A4 in GAP, Magma, Sage, TeX

Q_8\times A_4
% in TeX

G:=Group("Q8xA4");
// GroupNames label

G:=SmallGroup(96,199);
// by ID

G=gap.SmallGroup(96,199);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,2,72,169,79,376,665]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^2=d^2=e^3=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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Subgroup lattice of Q8×A4 in TeX
Character table of Q8×A4 in TeX

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