C₆₁×#13348 #29516781

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#13348

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂³×C₆₁
prime radical	C₂×C₆₁×B(4,𝔽₂)	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×D₄×C₆₁ ◁ G

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

normal subgroups only

This group has 900 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)(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14)(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);(5,7)(6,10)(8,17)(9,15)(11,20)(12,19)(13,16)(14,18);(2,4)(8,18)(11,16)

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, 6336);