Finite Group Databasebeta

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Cayley diagram

#2017⋊C3 #68032

order384
isoclinismisoclinic class of #2017⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#2017⋊C3

Splittings

Fitting, Sylow#2017 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4#127⋊C3
commutator, cosocle, hypocenter#338C2×C3
FrattiniC22×C4C2×A4
Fitting#2017C3
socleC2#392⋊C3
LCSG ▷ #338
UCSC1 ◁ C4

75 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:

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

Polycyclic presentation:

2,3,2^6|a^2=efg,c^2=fh,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=befh,c^a=ceg,d^a=defh,c^b=dg,d^b=cdefh,e^b=f,f^b=gh,g^b=eh,d^c=defgh,e^c=eh,g^c=gh,e^d=eh,f^d=fh

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#2017⋊C3, #23434⋊C3, #23794⋊C3, …
maximal normal subgroup ofC2×#2017⋊C3, #23434⋊C3, #23794⋊C3, …
maximal quotient of#11337⋊C3, C2×#2017⋊C3, C3×#2017⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 608);