C₆₁×#10936 #29514369

order	15616
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₆₁ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₆₁×#10936

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₁	#120
commutator	C₂³	C₂×C₄²×C₆₁
Frattini, cosocle	C₂⁵	C₂³×C₆₁
prime radical	C₂⁵×C₆₁	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂⁵×C₆₁ ◁ G

1258 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 302 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:

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

Conjugacy classes & character table

3904 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 3924);