C₄×C₉×C₄⋊C₄ #10588932

order	576
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product	C₄×C₉×C₄⋊C₄
unambiguous semidirect product	C₄⋊(C₄²×C₉)

	subgroup	G/H
center	C₂²×C₄×C₉	C₂²
commutator	C₂	C₂×C₄²×C₉
Frattini, socle, prime radical, cosocle	C₂³×C₃	C₂³×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

291 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 219 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 576

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 814);