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Cayley diagram

Q8×C22⋊A4 #87540

order384
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C24

Aliases

direct productQ8×C22⋊A4

Splittings

Fitting, SylowC24×Q8 ⋊ C3

Subgroups

subgroupG/H
center, FrattiniC2C22×C22⋊A4
commutator, socle, cosocleC25C22×C3
FittingC24×Q8C3
prime radicalC2×C22⋊A4C22
hypercenterQ8C22⋊A4
hypocenterC24C3×Q8
LCSG ▷ C25 ▷ C24
UCSC1 ◁ C2 ◁ Q8

1114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 16 points:

(2,3,4)(6,7,8)(9,11,12,16)(10,13,14,15);(5,8)(6,7)(9,10,12,14)(11,15,16,13);(1,4)(2,3)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup ofSL(2,𝔽3)×#1020, SL(2,𝔽3)×#1020⋊C2
maximal subgroup of#32879⋊C3, #32882⋊C3, Q8×#120⋊C3, …
maximal normal subgroup of#32879⋊C3, #32882⋊C3, Q8×#120⋊C3, …
maximal quotient ofC4⋊C4×C22⋊A4, #11710681, Q8×C22⋊SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20116);