C₆₁×#13607 #29517040

order	15616
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#13607

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#2127
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	#124	C₂³×C₆₁
prime radical	C₆₁×#535	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#535 ◁ G

332 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,11,13,26)(4,15,16,31)(5,18,20,9)(6,10)(7,12,22,29)(8,17)(14,24,19,27)(21,28,23,30)(25,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,6,14)(2,7,10,21)(4,12,16,28)(5,19,20,13)(8,24,25,11)(9,22,18,23)(15,27,31,26)(17,29,32,30);(4,16)(7,21)(8,25)(12,28)(15,31)(17,32)(22,23)(29,30)

Conjugacy classes & character table

1525 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6595);