C₁₁×#8515 #14585708

order	2816
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₁×#8515

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₁	#379
commutator	C₄²	C₂²×C₄×C₁₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₁
prime radical	C₁₁×#3567	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂×C₄²×C₁₁ ◁ G

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

normal subgroups only

This group has 644 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)(12,13,16,23,19,14,25,29)(15,31,34,67,36,57,53,66)(17,39,37,49,42,65,54,60)(18,24,38,21,43,30,55,27)(20,44,46,71,26,56,45,33)(22,50,52,62,28,61,64,41)(32,68,35,51,70,75,72,63)(40,47,73,48,74,58,69,59);(12,14)(13,19)(15,26)(16,21)(17,39)(18,30)(20,36)(22,41)(23,38)(24,43)(25,27)(28,62)(29,55)(31,40)(32,45)(33,51)(34,48)(35,58)(37,64)(42,65)(44,66)(46,70)(47,72)(49,61)(50,60)(52,54)(53,59)(56,67)(57,74)(63,71)(68,73)(69,75);(12,15,18,35)(13,20,24,47)(14,26,30,58)(16,34,38,70)(17,40,41,33)(19,36,43,72)(21,48,23,46)(22,51,39,31)(25,53,55,32)(27,59,29,45)(28,63,65,57)(37,73,50,44)(42,74,62,71)(49,67,52,75)(54,69,61,56)(60,66,64,68)

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2816, 1503);