C₆₁×#7266 #29510699

order	15616
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#7266

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#329
commutator	C₄²	C₄²×C₆₁
Frattini, cosocle	C₂×#97	C₂²×C₆₁
prime radical	C₂³×C₆₁	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×#97×C₆₁ ◁ G

218 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 86 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 — 2 generators on 81 points:

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

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, 254);