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#2868⋊C₃ #73222

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₄

Aliases

unambiguous semidirect product

Splittings

#2868 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#127⋊C₃
commutator, cosocle	C₂×C₄²	C₂²×C₃
Frattini, socle	C₂³	C₂²×A₄
Fitting	#2868	C₃
prime radical	C₂×C₄²⋊C₃	C₂²
hypercenter	C₄	#127⋊C₃
hypocenter	C₄²	C₃×Q₈
LCS	G ▷ C₂×C₄² ▷ C₄²
UCS	C₁ ◁ C₂ ◁ C₄

138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 24

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#32800⋊C₃, #32801⋊C₃, #33162⋊C₃, …
maximal normal subgroup of	#32800⋊C₃, #32801⋊C₃, #33162⋊C₃, …
maximal quotient of	C₄⋊(#384⋊C₃), #14712⋊C₃, #15190⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5798);