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Cayley diagram

#1454⋊C3 #68013

order384
isoclinismisoclinic class of #1454⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C4

Aliases

unambiguous semidirect product#1454⋊C3

Splittings

Fitting, Sylow#1454 ⋊ C3
derivedC82 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC82C2×C3
FrattiniC42C2×A4
Fitting#1454C3
socleC22C2×C42⋊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:

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

Polycyclic presentation:

2,3,2^6|c^2=e,d^2=f,e^2=g,f^2=h,b^a=bh,c^a=cgh,d^a=dg,c^b=dfh,d^b=cdfh,e^b=fh,f^b=efh,g^b=h,h^b=gh

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#1454⋊C3, #17339⋊C3, #19733⋊C3, …
maximal normal subgroup ofC2×#1454⋊C3, #17339⋊C3, #19733⋊C3, …
maximal quotient of#7845⋊C3, #7851⋊C3, #8135⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 589);