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Cayley diagram

#2645⋊C3 #73195

order384
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#2645⋊C3

Splittings

Fitting, Sylow#2645 ⋊ C3

Subgroups

subgroupG/H
centerC2C2×#126⋊C3
commutator, cosocle, hypocenter#95C22×C3
Frattini, socleC23C22×A4
Fitting#2645C3
prime radical#95⋊C3C22
hypercenterC4#126⋊C3
LCSG ▷ #95
UCSC1 ◁ C2 ◁ C4

87 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2^2,3,2^5|a^2=f,b^2=f,d^2=g,e^2=fgh,b^a=bf,c^a=cgh,d^a=dgh,e^a=efh,d^c=eh,e^c=defg,g^c=fgh,h^c=g,e^d=ef

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#32803⋊C3, #32805⋊C3, #32868⋊C3, …
maximal normal subgroup of#32803⋊C3, #32805⋊C3, #32868⋊C3, …
maximal quotient of#14333⋊C3, #14350⋊C3, #14909⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5771);