C₆₁×#7405 #29510838

order	15616
nilpotency class	5
isoclinism	isoclinic class of #7402
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#7405

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#353
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

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

normal subgroups only

This group has 252 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)(9,10,12,17,14,19,26,36)(11,20,21,34,23,31,38,39)(13,18,28,35,29,16,25,33)(15,24,30,27,32,37,40,22)(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)(9,11,14,23)(10,15,19,32)(12,24,26,37)(13,27,29,22)(16,31,18,20)(17,34,36,39)(21,25,38,28)(30,35,40,33)

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, 393);