C₁₁×#12466 #14589659

order	2816
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₁×#12466

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₁	#1677
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₁
prime radical	C₁₁×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₁ ◁ C₂×C₄×C₁₁ ◁ C₄×C₈×C₁₁ ◁ G

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

normal subgroups only

This group has 186 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.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 704

A minimal faithful permutation representation — 3 generators on 75 points:

(1,2,3,4,5,6,7,8,9,10,11)(12,13,20,27)(14,28,35,60)(15,36,40,52)(16,24,41,46)(17,23,29,55)(18,26,19,25)(21,45,51,31)(22,37,54,38)(30,62,65,74)(32,49,42,50)(33,48,34,56)(39,69,44,57)(43,58,61,53)(47,68,72,64)(59,66,71,63)(67,73,70,75);(12,14,29,32,18,33,16,31,20,35,17,42,19,34,41,45)(13,21,46,48,25,49,23,28,27,51,24,56,26,50,55,60)(15,30,61,67,39,64,38,66,40,65,43,70,44,68,37,63)(22,47,69,73,53,62,36,59,54,72,57,75,58,74,52,71);(12,15)(13,22)(14,30)(16,38)(17,43)(18,39)(19,44)(20,40)(21,47)(23,36)(24,57)(25,53)(26,58)(27,54)(28,59)(29,61)(31,66)(32,67)(33,64)(34,68)(35,65)(37,41)(42,70)(45,63)(46,69)(48,73)(49,62)(50,74)(51,72)(52,55)(56,75)(60,71)

Conjugacy classes & character table

341 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2816, 5454);