C₆₁×#7420 #29510853

order	15616
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#7420

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#357
commutator	C₂×C₁₆	C₂×C₄×C₆₁
Frattini, cosocle	C₂²×C₁₆	C₂²×C₆₁
prime radical	C₆₁×#2215	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ C₂²×C₁₆×C₆₁ ◁ G

150 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 50 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)(3,7,8,5)(4,6)(9,10,12,19,16,22,15,33)(11,23,26,39,29,17,28,34)(13,31,25,21,14,32,30,20)(18,37,35,24,38,40,36,27)(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)(2,5)(4,8)(6,7)(10,17)(11,24)(12,15)(13,30)(14,25)(18,21)(19,39)(20,38)(22,23)(26,37)(27,29)(28,40)(31,35)(32,36)(33,34)

Conjugacy classes & character table

2989 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 408);