C₂²×C₁₁×#123 #13066272

order	1408
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₁₁ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²×C₁₁×#123

	subgroup	G/H
center, socle	C₂⁴×C₁₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₁
prime radical	C₂³×D₄×C₁₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁ ◁ G

1968 conjugacy classes of subgroups; only the 952 normal subgroups are shown.

normal subgroups only

This group has 1968 conjugacy classes of subgroups — too many to draw. Only its 952 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) = 704

A minimal faithful permutation representation — 5 generators on 27 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1408, 2166);