C₆₁×#9966 #29513399

order	15616
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₁×#9966

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₁	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
prime radical	C₂³×D₄×C₆₁	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

1020 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1020 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) = 81
transitive degree	μ_t(G) = 3904

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

(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,14)(15,17)(16,18)(19,20)(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);(6,10)(7,11)(8,12)(17,20);(2,4)(5,7,9,6)(8,10,12,11)(13,15)(14,17)(16,19)(18,20)

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, 2954);