C₂²×C₆₁×#458 #29563826

order	15616
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₆₁ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×C₆₁×#458

Subgroups

	subgroup	G/H
center, socle	C₂³×C₆₁	#120
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₁
prime radical	C₂²×H(5,𝔽₂)×C₆₁	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂⁵×C₆₁ ◁ G

5294 conjugacy classes of subgroups; only the 1134 normal subgroups are shown.

normal subgroups only

This group has 5294 conjugacy classes of subgroups — too many to draw. Only its 1134 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) = 81
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 5 generators on 81 points:

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

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, 53381);