C₆₁×#10285 #29513718

order	15616
nilpotency class	3
isoclinism	isoclinic class of #10280
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₆₁×#10285

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₆₁×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

542 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 85
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 85 points:

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

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, 3273);