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Cayley diagram

#1010⋊C3 #64278

order288
automorphismAut(G) = D3×D4×S4 (order 1152) · Inn(G) = C2×D3×A4 · Out(G) = C23
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#1010⋊C3

Splittings

FittingC3×#41 ⋊ C2×C3

Subgroups

subgroupG/H
center, FrattiniC2C2×D3×A4
commutator, cosocle, hypocenterC3×Q8C22×C3
FittingC3×#41C2×C3
socleC2×C3C22×A4
hypercenterC4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2 ◁ C4

97 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

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

Polycyclic presentation:

2^2,3,2^2,3,2|b^2=g,d^2=g,e^2=g,b^a=bg,f^a=f^2,d^c=de,e^c=d,e^d=eg

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593229, #10593230, #10593231, …
maximal normal subgroup of#10593229, #10593230, #10593231, …
maximal quotient of#5570⋊C3, SL(2,𝔽3)×D12, #5572⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 926);