C₆₁×#7496 #29510929

order	15616
nilpotency class	5
isoclinism	isoclinic class of #7493
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#7496

Subgroups

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

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

normal subgroups only

This group has 274 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:

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

Conjugacy classes & character table

1708 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 484);