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Cayley diagram

C8×#41⋊C3 #73251

order384
isoclinismisoclinic class of SL(2,𝔽3)
automorphism?
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC8×#41⋊C3
unambiguous semidirect product(C8×#41)⋊C3

Splittings

Fitting, SylowC8×#41 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4×C8A4
commutator, hypocenterQ8C2×C3×C8
FrattiniC2×C4C22×A4
FittingC8×#41C3
socleC22C2×C4×A4
prime radicalC2×#41⋊C3C4
cosocleC4×Q8C22×C3
LCSG ▷ Q8
UCSC1 ◁ C4×C8

131 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:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×#131⋊C3, #17316⋊C3, #17320⋊C3, …
maximal normal subgroup ofC8×#131⋊C3, #17316⋊C3, #17320⋊C3, …
maximal quotient ofC4×C8×SL(2,𝔽3), #22530⋊C3, #22734⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5827);