C₆₁×#12375 #29515808

order	15616
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#12375

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₆₁
prime radical	C₆₁×#3283	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ C₈×D₄×C₆₁ ◁ G

350 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 86 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 — 3 generators on 101 points:

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

Conjugacy classes & character table

2440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5363);