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Cayley diagram

#519⋊C3 #5439

order192
automorphismAut(G) = S4×#120 (order 768) · Inn(G) = C22×A4 · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#519⋊C3

Splittings

Fitting, Sylow#519 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, socleC22C22×A4
commutator, hypocenterQ8C2×C3×C4
Fitting#519C3
prime radicalC22×SL(2,𝔽3)C2
cosocleC2×Q8C22×C3
hypercenterC22⋊C4A4
LCSG ▷ Q8
UCSC1 ◁ C22 ◁ C22⋊C4

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 20 points:

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

Polycyclic presentation:

2^2,3,2^4|b^2=d,e^2=g,f^2=g,b^a=bg,e^c=f,f^c=ef,f^e=fg

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#73074, #2884⋊C3, #85350, …
maximal normal subgroup of#73074, #2884⋊C3, #85350, …
maximal quotient of#73167, C22⋊C4×SL(2,𝔽3), #2351⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 998);