C₂×C₃²×#393 #11769434

order	1152
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁷×C₃

Aliases

C₂×C₃²×#393

	subgroup	G/H
center, socle	C₂⁴×C₃²	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃²
prime radical	C₂⁵×C₃²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃² ◁ G

3108 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 4 generators on 24 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3445);