C₆₁×#8322 #29511755

order	15616
nilpotency class	3
isoclinism	isoclinic class of #1795
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#8322

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#379
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂³×D₄×C₆₁	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

1146 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 214 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) = 1952

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

(1,2,3,5,4,6,7,8)(9,10)(11,18,20,15)(12,14)(13,17)(16,21)(19,23,24,22)(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);(10,14)(12,21)(15,22)(16,17)(18,23)(19,24);(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,18)(21,24)

Conjugacy classes & character table

3538 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 1310);