C₆₁×#7506 #29510939

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

Aliases

direct product

C₆₁×#7506

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#353
commutator	C₄²	C₄²×C₆₁
Frattini, cosocle	#532	C₂²×C₆₁
socle	C₂×C₆₁	#1393
prime radical	C₆₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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) = 93
transitive degree	μ_t(G) = 1952

A minimal faithful permutation representation — 2 generators on 93 points:

(1,2,4,9)(3,12,14,28)(5,8,19,24)(6,11,17,26)(7,16,21,15)(10,18,27,20)(13,23,29,25)(22,31,32,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3)(2,7)(4,16,17,31)(5,15)(6,13)(8,23)(9,25,26,28)(10,12)(11,22)(14,19,29,18)(20,30)(21,24,32,27)

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, 494);