C₁₁×#13568 #14590761

order	2816
nilpotency class	5
isoclinism	isoclinic class of #1404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₁×#13568

Subgroups

	subgroup	G/H
center	C₄×C₁₁	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₁₁
Frattini, cosocle	#124	C₂³×C₁₁
socle	C₂×C₁₁	C₂×AΣL(1,𝔽₁₆)
prime radical	C₁₁×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₁ ◁ C₂×C₄×C₁₁ ◁ C₂²×C₄×C₁₁ ◁ C₁₁×#532 ◁ G

498 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 90 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)(12,13,15,19)(14,17,22,24)(16,21,26,27)(18,23,20,25);(12,13)(14,17)(15,24)(16,21)(18,26)(19,25)(20,23)(22,27);(12,14,16,23)(13,17,21,20)(15,22,26,25)(18,19,24,27)

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, 6556);