C₆₁×#13357 #29516790

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13357
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₁×#13357

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂³×C₆₁
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×Q₈×C₆₁ ◁ G

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

normal subgroups only

This group has 772 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) = 81
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 3 generators on 81 points:

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

Conjugacy classes & character table

1891 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6345);