C₆₁×#13368 #29516801

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#13368

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂³×C₆₁	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×#97×C₆₁ ◁ G

370 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 101 points:

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

Conjugacy classes & character table

2074 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 6356);