C₁₁×#12302 #14589495

order	2816
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₁×#12302

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₁	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₁
prime radical	C₂×C₁₁×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₁ ◁ C₂×C₄×C₁₁ ◁ C₂×C₁₁×M₁₆ ◁ G

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

normal subgroups only

This group has 310 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)(12,13,18,23,19,25,17,24)(14,26,30,47,31,55,29,54)(15,32,34,21,35,50,40,46)(16,36,38,22,39,51,41,49)(20,37,52,28,44,60,43,61)(27,56,58,53,59,74,62,71)(33,48,67,64,68,45,66,69)(42,70,63,65,73,57,72,75);(12,14,19,31)(13,20,25,44)(15,27,35,59)(16,37,39,60)(17,30,18,29)(21,42,46,73)(22,47,49,54)(23,43,24,52)(26,36,55,51)(28,38,61,41)(32,63,50,72)(33,65,68,75)(34,62,40,58)(45,56,48,74)(53,69,71,64)(57,67,70,66);(12,15)(13,21)(14,27)(16,33)(17,40)(18,34)(19,35)(20,42)(22,48)(23,50)(24,32)(25,46)(26,53)(28,57)(29,62)(30,58)(31,59)(36,69)(37,65)(38,67)(39,68)(41,66)(43,72)(44,73)(45,49)(47,74)(51,64)(52,63)(54,56)(55,71)(60,75)(61,70)

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, 5290);