Finite Group Databasebeta

← Prev (#73268) Next (#73270) →

Cayley diagram

Q16×SL(2,𝔽3) #73269

order384
isoclinismisoclinic class of #3358⋊C3
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productQ16×SL(2,𝔽3)
unambiguous semidirect product(Q8×Q16)⋊C3

Splittings

Fitting, SylowQ8×Q16 ⋊ C3

Subgroups

subgroupG/H
center, socleC22D4×A4
commutator, cosocleC4×Q8C22×C3
FrattiniC2×C4C22×A4
FittingQ8×Q16C3
prime radicalC2×SL(2,𝔽3)D4
hypercenterC2×Q16A4
hypocenterQ8C3×Q16
LCSG ▷ C4×Q8 ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#33929⋊C3, #33930⋊C3, SL(2,𝔽3)×SD32, …
maximal normal subgroup of#33929⋊C3, #33930⋊C3, SL(2,𝔽3)×SD32, …
maximal quotient ofSL(2,𝔽3)×#103, SL(2,𝔽3)×#107, C2×Q16×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5845);