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Cayley diagram

#577⋊C3 #5459

order192
automorphismAut(G) = C2×D4×S4 (order 384) · Inn(G) = D4×A4 · Out(G) = C22
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#577⋊C3

Splittings

Fitting, Sylow#577 ⋊ C3

Subgroups

subgroupG/H
center, socleC2D4×A4
commutator, cosocle#41C22×C3
FrattiniC4C22×A4
Fitting#577C3
hypercenterSD16A4
hypocenterQ8C3×D4
LCSG ▷ #41 ▷ Q8
UCSC1 ◁ C2 ◁ C4 ◁ SD16

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:

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

Polycyclic presentation:

2^2,3,2^4|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#85469, #85470, #85471, …
maximal normal subgroup of#85469, #85470, #85471, …
maximal quotient of#2897⋊C3, #2898⋊C3, #2912⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1018);