C₆₁×#7413 #29510846

order	15616
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#7413

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#354
commutator	C₂×C₄²	C₂×C₄×C₆₁
Frattini, cosocle	#391	C₂²×C₆₁
prime radical	C₂³×C₆₁	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#391 ◁ G

178 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 46 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)(9,10,12,17,14,19,25,34)(11,20,22,37,21,35,38,33)(13,16,28,32,29,18,26,36)(15,24,30,27,31,39,40,23)(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);(2,5)(3,8)(6,7)(9,11)(10,15,19,31)(12,24)(13,27)(14,21)(16,20,18,35)(17,33,34,37)(22,28)(23,29)(25,39)(26,38)(30,36,40,32)

Conjugacy classes & character table

1708 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 401);