C₆₁×#9760 #29513193

order	15616
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂²×C₈

Aliases

direct product

C₆₁×#9760

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#390
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₆₁×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

478 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 101
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 101 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,15,20,30)(12,22)(13,27)(14,17)(16,25)(18,24)(19,33,23,37)(21,36,34,29)(26,38)(28,39,31,35)(32,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(1,3,7,2,5,8,4,6)(9,11,14,20)(10,15)(12,23,26,19)(13,21,24,34)(16,31)(17,30)(18,29)(22,37)(25,39,40,35)(27,36)(28,32)(33,38);(2,6)(3,8)(9,12,24,40,14,26,13,25)(10,16,27,38,17,32,18,22)(11,19,34,39,20,23,21,35)(15,28,36,37,30,31,29,33)

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, 2748);