C₆₁×#29710 #29533143

order	15616
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₁ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₆₁×#29710

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₆₁
prime radical	C₆₁×#3459	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

1116 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 242 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 — 4 generators on 85 points:

(2,3)(6,7)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(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);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23);(1,3,5,2)(4,6,8,7)(9,12,13,19)(10,15,16,23)(11,17,18,24)(14,20,21,22);(1,4,5,8)(2,6,3,7)

Conjugacy classes & character table

2257 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 22698);