C₆₁×#7486 #29510919

order	15616
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₁×#7486

Subgroups

	subgroup	G/H
center	C₄×C₆₁	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₄²×C₆₁
Frattini, cosocle	#532	C₂²×C₆₁
socle	C₂×C₆₁	#1390
prime radical	C₆₁×#532	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#532 ◁ G

286 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 77
transitive degree	μ_t(G) = 976

A minimal faithful permutation representation — 2 generators on 77 points:

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

Conjugacy classes & character table

2074 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 474);