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Cayley diagram

#1747⋊C3 #68023

order384
isoclinismisoclinic class of #338⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#1747⋊C3

Splittings

Fitting, Sylow#1747 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC8C42⋊C3
commutator, cosocle, hypocenter#338C2×C3
FrattiniC22×C4C2×A4
Fitting#1747C3
socleC2#376⋊C3
prime radical#338⋊C3C2
LCSG ▷ #338
UCSC1 ◁ C8

50 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:

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

Polycyclic presentation:

2,3,2^6|a^2=e,c^2=ef,d^2=eg,e^2=h,c^b=cdf,d^b=cefh,f^b=fg,g^b=fh,d^c=de,g^c=gh,f^d=fh

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#12542⋊C3, #12902⋊C3, C2×#1747⋊C3, …
maximal normal subgroup of#12542⋊C3, #12902⋊C3, C2×#1747⋊C3, …
maximal quotient of#10699⋊C3, C2×#1747⋊C3, C3×#1747⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 599);