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Cayley diagram

#1711⋊C3 #68018

order384
isoclinismisoclinic class of #1711⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C4

Aliases

unambiguous semidirect product#1711⋊C3

Splittings

Fitting, Sylow#1711 ⋊ C3
derivedC82 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC82C2×C3
FrattiniC42C2×A4
Fitting#1711C3
socleC22#127⋊C3
LCSG ▷ C82
UCSC1

43 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:

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

Polycyclic presentation:

2,3,2^6|c^2=egh,d^2=fg,e^2=g,f^2=h,b^a=beh,c^a=ce,d^a=df,e^a=eg,f^a=fh,c^b=cdefgh,d^b=cg,e^b=efgh,f^b=e,g^b=gh,h^b=g

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#1711⋊C3, #19733⋊C3, #19743⋊C3, …
maximal normal subgroup ofC2×#1711⋊C3, #19733⋊C3, #19743⋊C3, …
maximal quotient of#8817⋊C3, #10319⋊C3, #10678⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 594);