H(3,𝔽₃)×C₄⋊Q₈ #11721920

order	864
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)×#120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃²

Aliases

direct product	H(3,𝔽₃)×C₄⋊Q₈
unambiguous semidirect product	C₄⋊(Q₈×H(3,𝔽₃))
unambiguous semidirect product	(C₃²×C₄⋊Q₈)⋊C₃

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

374 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 1887);