C₆₁×#7324 #29510757

order	15616
nilpotency class	4
isoclinism	isoclinic class of #7304
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#7324

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#342
commutator	C₂⁴	C₄²×C₆₁
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂²×C₆₁
prime radical	C₂⁵×C₆₁	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂⁴×C₆₁ ◁ C₂²×C₂²⋊C₄×C₆₁ ◁ G

482 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 77
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 2 generators on 77 points:

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

Conjugacy classes & character table

2074 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 312);