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Cayley diagram

#2823⋊C3 #73206

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

Aliases

unambiguous semidirect product#2823⋊C3

Splittings

Fitting, Sylow#2823 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#560⋊C3
commutator, cosocleC2×C42C22×C3
Frattini, socleC23C22×A4
Fitting#2823C3
hypocenterC42C3×D4
LCSG ▷ C2×C42 ▷ C42
UCSC1 ◁ C2

116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 20 points:

(2,4)(5,6,7)(8,18,17,16,19,14)(9,11,15,12,13,20);(1,2)(3,4)(5,15,7,20)(6,19,10,18)(8,17,13,12)(9,11,14,16)

Polycyclic presentation:

2^2,3,2^5|e^2=g,f^2=h,b^a=bd,c^a=cg,e^a=egh,f^a=fg,c^b=ch,e^b=eh,f^b=fgh,e^c=ef,f^c=eh,g^c=gh,h^c=g

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#2823⋊C3, #36696⋊C3, #45970⋊C3, …
maximal normal subgroup ofC2×#2823⋊C3, #36696⋊C3, #45970⋊C3, …
maximal quotient of#14665⋊C3, #14676⋊C3, #17135⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5782);