C₆₁×#10498 #29513931

order	15616
nilpotency class	3
isoclinism	isoclinic class of #10496
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#10498

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂²×D₄×C₆₁	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

380 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 90 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)(4,7)(6,8)(9,10,14,18)(11,12,16,22)(13,19,24,17)(15,23,21,20)(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);(3,8)(5,6)(9,11)(10,15)(12,19)(13,23)(14,16)(17,22)(18,21)(20,24);(2,5)(3,8)(6,7)(9,12,13,21,14,22,24,15)(10,16,17,23,18,11,19,20)

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, 3486);