C₆₁×#12777 #29516210

order	15616
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#12777

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×Q₈	C₂³×C₆₁
prime radical	C₂³×C₆₁	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×Q₈×C₆₁ ◁ G

784 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 1952

A minimal faithful permutation representation — 3 generators on 81 points:

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

Conjugacy classes & character table

2440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5765);