C₁₁×#7391 #14584584

order	2816
nilpotency class	5
isoclinism	isoclinic class of #7389
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₁×#7391

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₁	#1333
commutator	C₁₆	C₂×C₈×C₁₁
Frattini, cosocle	#346	C₂²×C₁₁
prime radical	C₂²×C₁₁	#361
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₁ ◁ C₂×C₄×C₁₁ ◁ C₄×C₈×C₁₁ ◁ C₁₁×#346 ◁ G

116 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 75 points:

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

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, 379);