#2275⋊C₃ #73165

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

Aliases

	subgroup	G/H
center	C₂	C₂²×C₄²⋊C₃
commutator, cosocle, hypocenter	#95	C₂²×C₃
Frattini, socle	C₂³	C₂²×A₄
Fitting	#2275	C₃
hypercenter	D₄	C₄²⋊C₃
LCS	G ▷ #95
UCS	C₁ ◁ C₂ ◁ D₄

165 conjugacy classes of subgroups.

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

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

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

Polycyclic presentation:

2^2,3,2^5|d^2=h,e^2=g,b^a=bf,c^a=cgh,c^b=cfg,d^c=deg,e^c=dfgh,g^c=h,h^c=fgh,e^d=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5741);