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#2843⋊C₃ #73213

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

Aliases

unambiguous semidirect product

Splittings

#2843 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#560⋊C₃
commutator, cosocle	C₂×C₄²	C₂²×C₃
Frattini, socle	C₂³	C₂²×A₄
Fitting	#2843	C₃
prime radical	C₂×C₄²⋊C₃	C₂²
hypocenter	C₄²	C₃×Q₈
LCS	G ▷ C₂×C₄² ▷ C₄²
UCS	C₁ ◁ C₂

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 48

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

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

Polycyclic presentation:

2^2,3,2^5|a^2=d,b^2=d,e^2=h,f^2=g,b^a=bd,e^a=eg,f^a=fgh,c^b=cgh,e^b=eh,f^b=fg,e^c=efgh,f^c=e,g^c=h,h^c=gh

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#2843⋊C₃, #36696⋊C₃, #45982⋊C₃, …
maximal normal subgroup of	C₂×#2843⋊C₃, #36696⋊C₃, #45982⋊C₃, …
maximal quotient of	#14676⋊C₃, #14678⋊C₃, #17114⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5789);