C₁₁×#13198 #14590391

order	2816
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₁×#13198

Subgroups

	subgroup	G/H
center	C₈×C₁₁	#127
commutator	C₂²×C₄	C₂²×C₄×C₁₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₁
socle	C₂×C₁₁	#1955
prime radical	C₁₁×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₁ ◁ C₂²×C₈×C₁₁ ◁ G

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

normal subgroups only

This group has 410 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) = 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)(12,13,16,21,17,22,32,43,18,23,33,44,34,45,58,69)(14,19,26,37,27,38,52,64,25,36,50,62,51,63,73,75)(15,20,30,41,31,42,47,60,29,40,54,66,57,68,72,70)(24,46,49,71,28,53,56,74,48,35,59,61,55,39,65,67);(12,14,16,26,17,27,32,52,18,25,33,50,34,51,58,73)(13,19,21,37,22,38,43,64,23,36,44,62,45,63,69,75)(15,28,54,65,31,48,72,49,29,55,30,56,57,24,47,59)(20,39,66,74,42,46,70,61,40,53,41,67,68,35,60,71);(12,15)(13,20)(14,24,25,48)(16,30)(17,31)(18,29)(19,35,36,46)(21,41)(22,42)(23,40)(26,49,50,59)(27,28,51,55)(32,47)(33,54)(34,57)(37,61,62,71)(38,39,63,53)(43,60)(44,66)(45,68)(52,56,73,65)(58,72)(64,67,75,74)(69,70)

Conjugacy classes & character table

572 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2816, 6186);