C₆₁×#13497 #29516930

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13497
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#13497

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#2075
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂×#41×C₆₁	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×#131×C₆₁ ◁ G

274 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Conjugacy classes & character table

2074 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6485);