C₆₁×#11338 #29514771

order	15616
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₁×#11338

Subgroups

	subgroup	G/H
center	C₂×C₄×C₆₁	#127
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
socle	C₂²×C₆₁	#399
prime radical	C₂⁴×C₆₁	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

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

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 254 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 — 3 generators on 81 points:

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

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, 4326);