C₁₁×#12305 #14589498

order	2816
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₁×#12305

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₁	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₁
prime radical	C₂³×C₁₁	C₂×SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₁ ◁ C₂×C₄×C₁₁ ◁ C₂×C₁₁×M₁₆ ◁ G

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

normal subgroups only

This group has 262 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)(12,13,17,37,18,23,39,22)(14,24,28,46,27,51,38,52)(15,29,32,45,31,20,43,59)(16,33,36,21,35,65,66,48)(19,34,49,26,42,56,41,57)(25,50,54,74,55,68,64,71)(30,60,62,47,63,72,70,44)(40,67,73,75,58,61,69,53);(12,14,18,27)(13,19,23,42)(15,25,31,55)(16,34,35,56)(17,38,39,28)(20,40,29,58)(21,46,48,52)(22,49,37,41)(24,33,51,65)(26,36,57,66)(30,61,63,67)(32,64,43,54)(44,71,47,74)(45,73,59,69)(50,60,68,72)(53,62,75,70);(12,15)(13,20,23,29)(14,25)(16,30)(17,32)(18,31)(19,40,42,58)(21,47,48,44)(22,45,37,59)(24,50,51,68)(26,53)(27,55)(28,54)(33,60,65,72)(34,61)(35,63)(36,62)(38,64)(39,43)(41,69,49,73)(46,74,52,71)(56,67)(57,75)(66,70)

Conjugacy classes & character table

275 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2816, 5293);