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Cayley diagram

#568⋊C3 #5454

order192
automorphismAut(G) = C2×D4×S4 (order 384) · Inn(G) = C22×A4 · Out(G) = C23
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#568⋊C3

Splittings

Fitting, Sylow#568 ⋊ C3

Subgroups

subgroupG/H
center, FrattiniC4C22×A4
commutator, hypocenterQ8C2×C3×C4
Fitting#568C3
socleC2C2×C4×A4
prime radicalC2×#41⋊C3C2
cosocle#41C22×C3
hypercenterM16A4
LCSG ▷ Q8
UCSC1 ◁ C4 ◁ M16

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:

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

Polycyclic presentation:

2^2,3,2^4|b^2=d,d^2=g,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#73129, #73130, #73131, …
maximal normal subgroup of#73129, #73130, #73131, …
maximal quotient of#2875⋊C3, #2880⋊C3, C4⋊(#131⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1013);