#2785⋊C₃ #73201

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

Aliases

	subgroup	G/H
center, hypercenter	C₂	#5450
commutator, cosocle, hypocenter	C₂²×Q₈	C₂²×C₃
Frattini, socle	C₂³	C₂²×A₄
Fitting	#2785	C₃
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂

108 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:

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

Polycyclic presentation:

2^2,3,2^5|b^2=h,d^2=h,e^2=h,b^a=bh,d^a=dfh,e^a=egh,d^b=dgh,e^b=efg,d^c=defg,e^c=dg,f^c=fgh,g^c=f,e^d=eh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5777);