C₆₁×#8550 #29511983

order	15616
nilpotency class	3
isoclinism	isoclinic class of #8541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#8550

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#379
commutator	C₂⁴	C₂²×C₄×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
prime radical	C₂⁵×C₆₁	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

1134 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 1952

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

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

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, 1538);