C₆₁×#11537 #29514970

order	15616
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#11537

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#410
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×M₁₆	C₂³×C₆₁
socle	C₂×C₆₁	#1466
prime radical	C₆₁×#525	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂×C₈×C₆₁ ◁ C₆₁×#405 ◁ G

348 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 93
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 3 generators on 93 points:

(1,2,4,9,5,10,17,25,6,7,16,20,18,21,28,32)(3,11,12,27,13,19,29,24,14,26,15,31,30,8,22,23)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(2,7)(8,19)(10,21)(11,26);(1,3,5,13,6,14,18,30)(2,8,21,26,7,19,10,11)(4,15,17,22,16,12,28,29)(9,24,32,27,20,23,25,31)

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, 4525);