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#2863⋊C₃ #73220

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

Splittings

#2863 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#126⋊C₃
commutator, cosocle	C₂×C₄²	C₂²×C₃
Frattini, socle	C₂³	C₂²×A₄
Fitting	#2863	C₃
prime radical	C₂×C₄²⋊C₃	C₂²
hypercenter	C₄	#126⋊C₃
hypocenter	C₄²	C₃×Q₈
LCS	G ▷ C₂×C₄² ▷ C₄²
UCS	C₁ ◁ C₂ ◁ C₄

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

2^2,3,2^5|a^2=d,b^2=d,e^2=gh,f^2=g,b^a=bd,c^a=cg,e^a=eg,f^a=fh,e^c=efg,f^c=eh,g^c=gh,h^c=g

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#32798⋊C₃, #32801⋊C₃, #32860⋊C₃, …
maximal normal subgroup of	#32798⋊C₃, #32801⋊C₃, #32860⋊C₃, …
maximal quotient of	C₄⋊(#383⋊C₃), #14710⋊C₃, #15189⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5796);