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Cayley diagram

Q8×A4 #992

order96
automorphismAut(G) = S42 (order 576) · Inn(G) = C22×A4 · Out(G) = C2×D3
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productQ8×A4

Splittings

Fitting, SylowC22×Q8 ⋊ C3

Subgroups

subgroupG/H
center, FrattiniC2C22×A4
commutator, socle, cosocleC23C22×C3
FittingC22×Q8C3
prime radicalC2×A4C22
hypercenterQ8A4
hypocenterC22C3×Q8
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2 ◁ Q8

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 12 points:

(2,3,4)(5,7,8,12)(6,9,10,11);(1,4)(2,3)(5,6,8,10)(7,11,12,9)

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

derived subgroup ofA4⋊(#41⋊C3), SL(2,𝔽3)×S4, SL(2,𝔽3)×SL(2,ℤ/4), …
maximal subgroup of#5416, #5417, A4×SD16, …
maximal normal subgroup of#5416, #5417, A4×SD16, …
maximal quotient ofA4×C4⋊C4, #557⋊C3, Q8×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 199);