C₆₁×#23794 #29527227

order	15616
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#23794

Subgroups

	subgroup	G/H
center	C₈×C₆₁	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle	C₂×C₆₁	#2397
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ G

930 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 250 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) = 93
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 4 generators on 93 points:

(1,2,5,9,6,10,18,24)(3,11,13,19,14,23,17,7)(4,8,15,21,16,22,29,32)(12,25,26,30,27,31,28,20)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3)(2,7)(4,12)(5,17)(6,14)(8,20)(9,23)(10,19)(11,24)(13,18)(15,28)(16,27)(21,31)(22,30)(25,32)(26,29);(7,19)(11,23)(20,30)(25,31);(1,4,5,15,6,16,18,29)(2,8,9,21,10,22,24,32)(3,12,17,28,14,27,13,26)(7,20,23,31,19,30,11,25)

Conjugacy classes & character table

3172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 16782);