C₆₁×#9411 #29512844

order	15616
nilpotency class	3
isoclinism	isoclinic class of #9385
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#9411

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#386
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
prime radical	C₂⁵×C₆₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

606 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 122 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:

(1,2,5,11,21,28,4,18)(3,13,17,26,31,32,15,27)(6,12,20,29,7,9,22,10)(8,16,25,23,24,14,30,19)(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)(2,8,12,26)(4,19)(5,23)(6,14)(7,16)(9,27,28,24)(10,25,18,32)(11,13,29,30)(15,20)(17,22)(21,31);(2,9)(3,14)(4,20)(5,22)(8,24)(12,28)(13,30)(15,17)(16,31)(19,23)(25,32)(26,27)

Conjugacy classes & character table

2257 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 2399);