C₆₁×#11394 #29514827

order	15616
nilpotency class	3
isoclinism	isoclinic class of #2032
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#11394

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#393
commutator	C₂⁴	C₂²×C₄×C₆₁
Frattini, cosocle	C₂⁵	C₂³×C₆₁
prime radical	C₂⁵×C₆₁	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂⁵×C₆₁ ◁ G

1090 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 1090 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 1952

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

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

Conjugacy classes & character table

2806 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 4382);