C₆₁×#13534 #29516967

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#13534

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂×#135×C₆₁	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

330 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 97
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 97 points:

(5,6,13,17,20,35,9,16,12,19,10,26,11,18,25,29)(7,15,22,32,28,30,21,31,24,34,8,14,23,33,27,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,2)(3,4)(5,7,20,23,12,24,11,28)(6,14,18,36,19,32,35,31)(8,13,21,9,22,10,27,25)(15,29,30,26,34,16,33,17);(2,4)(5,8)(6,15)(7,13)(9,23)(10,24)(11,27)(12,22)(14,29)(16,32)(17,31)(18,30)(19,34)(20,21)(25,28)(26,36)(33,35)

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, 6522);