C₆₁×#13533 #29516966

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#13533

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂×#135×C₆₁	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

346 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 81
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 81 points:

(5,6,9,16,15,20,8,7,11,14,19,12,10,13,18,17)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(1,2)(3,4)(5,7,15,16,11,17,10,12)(6,9,13,8,14,19,20,18);(2,4)(5,6)(7,9)(8,16)(10,20)(11,14)(12,18)(13,15)(17,19)

Conjugacy classes & character table

2440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6521);