C₃³×C₂²⋊Q₈ #11724003

order	864
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃³ · Schur multiplier (H₂) = C₂²×C₃³

Aliases

C₃³×C₂²⋊Q₈

	subgroup	G/H
center, socle	C₂²×C₃³	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃³
prime radical	C₂³×C₃³	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃³ ◁ G

1036 conjugacy classes of subgroups; only the 672 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 672 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) = 432

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 3970);