C₆₁×#12736 #29516169

order	15616
nilpotency class	4
isoclinism	isoclinic class of #11663
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#12736

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂³×D₈×C₆₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

1162 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 182 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)(5,6,8,12,9,14,19,13)(7,15,17,11,18,10,16,20)(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)(7,16)(8,19)(11,12)(13,20)(14,15)(17,18);(2,4)

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, 5724);