C₆₁×#10564 #29513997

order	15616
nilpotency class	3
isoclinism	isoclinic class of #10564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₁×#10564

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#394
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
prime radical	C₆₁×#3186	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

574 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 77
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 77 points:

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

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, 3552);