C₁₁×#397 #10622884

order	704
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₁ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂³×C₁₁	C₂³
commutator, Frattini, cosocle	C₂³	C₂³×C₁₁
prime radical	C₂⁴×C₁₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ G

170 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

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

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 352

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3,4,5,6,7,8,9,10,11)(16,17,18,19)(21,24)(25,26);(13,15)(20,21)(22,26)(23,24)(25,27);(12,13)(14,15)(16,17)(18,19)(20,22,23,27)(21,25,24,26)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(704, 803);