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Cayley diagram

C24×SL(2,𝔽3) #87527

order384
isoclinismisoclinic class of SL(2,𝔽3)
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC24×SL(2,𝔽3)

Splittings

Fitting, SylowC24×Q8 ⋊ C3
derivedQ8 ⋊ C24×C3

Subgroups

subgroupG/H
center, socle, hypercenterC25A4
commutator, cosocle, hypocenterQ8C24×C3
FrattiniC2C24×A4
FittingC24×Q8C3
LCSG ▷ Q8
UCSC1 ◁ C25

1779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

derived subgroup ofGL(2,𝔽3)×C22⋊A4, #72241465, GL(2,𝔽3)×#120⋊C3
maximal subgroup ofC2×C22⋊C4×SL(2,𝔽3), #34173⋊C3, SL(2,𝔽3)×#120, …
maximal normal subgroup ofC2×C22⋊C4×SL(2,𝔽3), #34173⋊C3, SL(2,𝔽3)×#120, …
maximal quotient ofC23×C4×SL(2,𝔽3), C22×D4×SL(2,𝔽3), C22×Q8×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20103);