D₄×C₃²⋊C₉ #10619696

order	648
nilpotency class	2
isoclinism	isoclinic class of D₄×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₉ · Schur multiplier (H₂) = C₂×C₃²

Aliases

direct product	D₄×C₃²⋊C₉
unambiguous semidirect product	C₄⋊(C₂×C₃²⋊C₉)
unambiguous semidirect product	(C₃²×D₄)⋊C₉

	subgroup	G/H
center, Frattini, socle, cosocle	C₂×C₃²	C₂²×C₃²
commutator	C₂×C₃	C₂²×C₃×C₉
prime radical	C₃³×D₄	C₃
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ G

184 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 2 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 215);