C₁₁×#7413 #14584606

order	2816
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₁×#7413

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₁	#354
commutator	C₂×C₄²	C₂×C₄×C₁₁
Frattini, cosocle	#391	C₂²×C₁₁
prime radical	C₂³×C₁₁	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂³×C₁₁ ◁ C₂²×C₄×C₁₁ ◁ C₁₁×#391 ◁ G

178 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 51
transitive degree	μ_t(G) = 704

A minimal faithful permutation representation — 2 generators on 51 points:

(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,16)(15,18)(17,19)(20,21,23,28,25,30,36,45)(22,31,33,48,32,46,49,44)(24,27,39,43,40,29,37,47)(26,35,41,38,42,50,51,34);(13,16)(14,19)(17,18)(20,22)(21,26,30,42)(23,35)(24,38)(25,32)(27,31,29,46)(28,44,45,48)(33,39)(34,40)(36,50)(37,49)(41,47,51,43)

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2816, 401);