C₁₁×#12722 #14589915

order	2816
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₁×#12722

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₁	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₁
Frattini, cosocle	C₂×#41	C₂³×C₁₁
prime radical	C₁₁×#3508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₁ ◁ C₂×C₄×C₁₁ ◁ C₄×C₁₁×#41 ◁ G

790 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 176

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

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

Conjugacy classes & character table

374 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2816, 5710);