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Cayley diagram

#578⋊C3 #5460

order192
automorphismAut(G) = S4×AGL(1,ℤ/8) (order 768) · Inn(G) = D4×A4 · Out(G) = C23
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#578⋊C3

Splittings

Fitting, Sylow#578 ⋊ C3

Subgroups

subgroupG/H
center, socleC2D4×A4
commutator, cosocle#41C22×C3
FrattiniC4C22×A4
Fitting#578C3
hypercenterD8A4
hypocenterQ8C3×D4
LCSG ▷ #41 ▷ Q8
UCSC1 ◁ C2 ◁ C4 ◁ D8

73 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#73139, #73140, #3416⋊C3, …
maximal normal subgroup of#73139, #73140, #3416⋊C3, …
maximal quotient of#2898⋊C3, #2913⋊C3, #3359⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1019);