C₆₁×#7297 #29510730

order	15616
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#7297

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	#102
commutator	C₂×C₈	C₄²×C₆₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₁
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

208 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 7808

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

(1,2,5,7,6,3,4,8)(9,10,13,19,16,21,15,32)(11,22,24,34,27,36,26,40)(12,28,30,37,31,20,14,18)(17,25,33,29,35,39,38,23)(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);(1,3,6,2)(4,7,5,8)(9,11,14,25,15,26,31,23,16,27,30,39,13,24,12,29)(10,17,20,34,19,33,18,36,21,35,28,40,32,38,37,22)

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, 285);