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Cayley diagram

#2658⋊C3 #73198

order384
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C4

Aliases

unambiguous semidirect product#2658⋊C3

Splittings

Fitting, Sylow#2658 ⋊ C3

Subgroups

subgroupG/H
centerC2C2×#127⋊C3
commutator, cosocle, hypocenter#95C22×C3
Frattini, socleC23C22×A4
Fitting#2658C3
hypercenterC4#127⋊C3
LCSG ▷ #95
UCSC1 ◁ C2 ◁ C4

137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 28 points:

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

Polycyclic presentation:

2^2,3,2^5|b^2=fg,d^2=g,e^2=fgh,b^a=bfg,c^a=cf,d^a=df,e^a=eh,c^b=ch,d^c=eh,e^c=defgh,f^c=h,g^c=fgh,h^c=fh,e^d=efg

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#32804⋊C3, #32805⋊C3, #33168⋊C3, …
maximal normal subgroup of#32804⋊C3, #32805⋊C3, #33168⋊C3, …
maximal quotient of#14322⋊C3, #14354⋊C3, #14910⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5774);