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Cayley diagram

C3×#984 #64254

order288
automorphismAut(G) = C2×D4×S4 (order 384) · Inn(G) = C2×S4 · Out(G) = C23
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×#984

Subgroups

subgroupG/H
center, socleC2×C3C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C22×C3
FrattiniC2C2×C3×S4
FittingC3×#41D3
prime radicalC3×#41⋊C3C2
hypercenterC3×C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4

77 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 35 points:

(1,29,18,3,4,30,6,12,16,14,17,28)(2,25,24,7,8,32,10,19,22,11,23,31)(5,27,15,13)(9,26,21,20)(33,34,35);(1,10,6,2)(3,7,14,11)(4,21,17,9)(5,23,15,8)(12,20,29,26)(13,19,27,25)(16,22,18,24)(28,32,30,31)

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593224, #10593226, #10593231, …
maximal normal subgroup of#10593224, #10593226, #10593231, …
maximal quotient ofC3×#5386, C3×#5388, C3×C4⋊#226, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 902);