#73293

order	384
automorphism	Aut(G) = #11921467 (order 1152) · Inn(G) = C₂²⋊A₄ · Out(G) = S₄
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#2840 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂³	C₂²⋊A₄
commutator, Fitting, cosocle, hypocenter	#2840	C₃
LCS	G ▷ #2840
UCS	C₁ ◁ C₂³

185 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

3,2^7|c^2=h,d^2=h,b^a=be,c^a=cd,d^a=c,e^a=b,c^b=cf,d^b=dfg,d^c=dh,e^c=eg,e^d=ef

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11714249, #11714257, #72227539, …
maximal subgroup of	#10390⋊C₃, #10391⋊C₃, C₂×#73293, …
maximal normal subgroup of	#10390⋊C₃, #10391⋊C₃, C₂×#73293, …
maximal quotient of	#11709341, C₂×#73293, C₃×#73293, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5869);