C₆₁×#13543 #29516976

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#13543

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂³×C₆₁	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

268 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 85
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 85 points:

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

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, 6531);