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Cayley diagram

#576⋊C3 #5458

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#576⋊C3

Splittings

Fitting, Sylow#576 ⋊ C3

Subgroups

subgroupG/H
center, socleC2D4×A4
commutator, cosocle#41C22×C3
FrattiniC4C22×A4
Fitting#576C3
hypercenterQ16A4
hypocenterQ8C3×D4
LCSG ▷ #41 ▷ Q8
UCSC1 ◁ C2 ◁ C4 ◁ Q16

73 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 48 points:

(1,14,36,15,4,32,8,18,16,3,20,35)(2,24,44,26,10,40,12,41,27,9,30,48)(5,38,34,6,17,47,7,19,33,22,23,39)(11,29,46,13,28,43,21,31,42,25,37,45);(1,25)(2,3)(4,28)(5,26)(6,21)(7,9)(8,13)(10,17)(11,22)(12,15)(14,24)(16,40)(18,41)(19,29)(20,37)(23,30)(27,47)(31,38)(32,46)(33,43)(34,45)(35,42)(36,48)(39,44)

Polycyclic presentation:

2^2,3,2^4|a^2=g,b^2=g,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#73137, #73138, #3415⋊C3, …
maximal normal subgroup of#73137, #73138, #3415⋊C3, …
maximal quotient of#2897⋊C3, #2913⋊C3, #3362⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1017);