C₆₁×#7261 #29510694

order	15616
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#7261

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	#102
commutator	C₂×C₈	C₄²×C₆₁
Frattini, cosocle	C₂³×C₈	C₂²×C₆₁
socle	C₂²×C₆₁	#340
prime radical	C₂⁴×C₆₁	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

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

normal subgroups only

This group has 320 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) = 101
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 2 generators on 101 points:

(1,2)(3,5)(4,7)(6,8)(9,10,13,18,15,19,14,29)(11,20,22,36,23,37,30,34)(12,24,26,35,28,39,27,17)(16,25,31,38,33,40,32,21)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(2,5,7,6)(9,11,13,22,15,23,14,30)(10,16,18,31,19,33,29,32)(12,25,26,38,28,40,27,21)(17,34,24,20,35,36,39,37)

Conjugacy classes & character table

3538 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 249);