C₂×H(3,𝔽₃)×C₂²⋊Q₈ #13154552

order	1728
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃²

Aliases

direct product	C₂×H(3,𝔽₃)×C₂²⋊Q₈
unambiguous semidirect product	(C₂×C₃²×C₂²⋊Q₈)⋊C₃

	subgroup	G/H
center, socle	C₂³×C₃	C₂³×C₃²
commutator, Frattini, cosocle	C₂²×C₃	C₂⁴×C₃²
prime radical	C₂⁴×H(3,𝔽₃)	C₂²
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

1771 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 20615);