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Cayley diagram

#557⋊C3 #5447

order192
isoclinismisoclinic class of #548⋊C3
automorphismAut(G) = S4×#120 (order 768) · Inn(G) = C22×A4 · Out(G) = C2×D4
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#557⋊C3

Splittings

Fitting, Sylow#557 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, socleC22C22×A4
commutator, cosocleC2×Q8C22×C3
Fitting#557C3
prime radicalC2×SL(2,𝔽3)C22
hypercenterC4⋊C4A4
hypocenterQ8C3×Q8
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ C4⋊C4

65 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#73093, #73094, #73095, …
maximal normal subgroup of#73093, #73094, #73095, …
maximal quotient ofC4⋊C4×SL(2,𝔽3), #2351⋊C3, C2×#557⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1006);