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Cayley diagram

#1726⋊C3 #68021

order384
isoclinismisoclinic class of #95⋊C3
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#1726⋊C3

Splittings

Fitting, Sylow#1726 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC8C42⋊C3
commutator, hypocenter#95C3×C4
FrattiniC22×C4C2×A4
Fitting#1726C3
socleC23C4×A4
prime radical#95⋊C3C4
cosocle#376C2×C3
LCSG ▷ #95
UCSC1 ◁ C8

73 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 44
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 44 points:

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

Polycyclic presentation:

2,3,2^6|a^2=c,c^2=h,d^2=f,e^2=g,d^b=efh,e^b=deh,f^b=g,g^b=fgh,e^d=eh

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#12539⋊C3, #12895⋊C3, #12899⋊C3, …
maximal normal subgroup of#12539⋊C3, #12895⋊C3, #12899⋊C3, …
maximal quotient ofC8×#95⋊C3, #12542⋊C3, C2×#1726⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 597);