C₆₁×#10562 #29513995

order	15616
nilpotency class	3
isoclinism	isoclinic class of #10560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#10562

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#394
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
prime radical	C₂×#121×C₆₁	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

464 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 94 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:

(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,20,23)(12,15,13,17)(16,21,24,22)(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);(1,3)(2,5)(4,7)(6,8)(10,15)(11,20)(12,13)(16,19)(17,18)(23,24);(5,8)(9,11,12,21,14,20,13,22)(10,16,17,23,18,24,15,19)

Conjugacy classes & character table

1891 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 3550);