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Cayley diagram

C23×GL(2,𝔽3) #87479

order384
isoclinismisoclinic class of #226
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC23×GL(2,𝔽3)

Splittings

FittingC23×Q8 ⋊ D3
derivedSL(2,𝔽3) ⋊ C24

Subgroups

subgroupG/H
center, socle, hypercenterC24S4
commutator, cosocle, hypocenterSL(2,𝔽3)C24
FrattiniC2C23×S4
FittingC23×Q8D3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C24

1187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 14 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#73065, #11712730, #11712924, …
maximal normal subgroup ofC2×#73065, #11712730, #11712924, …
maximal quotient ofC23×#859, C22×C4×GL(2,𝔽3), C22×#5421, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20055);