C₆₁×#12964 #29516397

order	15616
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₆₁×#12964

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×D₄	C₂³×C₆₁
prime radical	C₂×#127×C₆₁	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×D₄×C₆₁ ◁ G

672 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 73
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 3 generators on 73 points:

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

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, 5952);