C₆₁×#8545 #29511978

order	15616
nilpotency class	3
isoclinism	isoclinic class of #8541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₆₁×#8545

Subgroups

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

910 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 93
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 3 generators on 93 points:

(1,2)(3,11,14,7)(4,16,17,8)(5,9)(6,10)(12,19)(13,27,28,20)(15,24,25,21)(18,32,26,22)(23,29,31,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,12,25)(2,7,19,24)(4,13,26,28)(5,14,6,15)(8,20,32,27)(9,11,10,21)(16,31,22,23)(17,30,18,29);(1,4,5,18)(2,8,9,22)(3,13,14,29)(6,17,12,26)(7,20,11,23)(10,16,19,32)(15,30,25,28)(21,31,24,27)

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, 1533);