C₆₁×#13542 #29516975

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#13542

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂³×C₆₁	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

252 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,5)(4,6)(7,8)(9,10,12,17,13,18,25,15)(11,20,22,29,23,16,24,32)(14,19,26,34,27,35,40,30)(21,37,38,31,39,33,28,36)(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);(1,3,4,7)(2,5,6,8)(10,15)(11,21,23,39)(12,25)(14,26)(16,31,20,36)(17,18)(22,28,24,38)(27,40)(29,33,32,37)(30,34);(2,6)(5,8)(9,11,12,22,13,23,25,24)(10,16,17,32,18,20,15,29)(14,28,26,21,27,38,40,39)(19,31,34,33,35,36,30,37)

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, 6530);