C₆₁×#13541 #29516974

order	15616
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#13541

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂³×C₆₁	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×#97×C₆₁ ◁ G

254 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 85
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 3 generators on 85 points:

(1,2,5,7)(3,8,4,6)(11,18)(14,19)(16,24)(20,21)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,5,4)(2,6,7,8)(9,10,12,22,13,15,23,17)(11,19,21,16,18,14,20,24);(1,4,5,3)(2,6,7,8)(10,14)(11,20)(12,23)(15,19)(16,22)(17,24)(18,21)

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, 6529);