C₆₁×#7115 #29510548

order	15616
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₆₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₆₁×#7115

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#329
commutator	C₄²	C₂×C₈×C₆₁
Frattini, cosocle	#376	C₂²×C₆₁
socle	C₂×C₆₁	#1270
prime radical	C₆₁×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#376 ◁ G

248 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,11,13,26,14,23,18,32)(4,15,16,30)(5,9)(6,10)(7,12,20,27,22,29,21,28)(8,17,24,31)(19,25)(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)(4,12)(5,18)(6,14)(8,23)(9,20)(10,22)(11,24)(13,19)(15,26)(16,29)(17,27)(21,25)(28,31)(30,32)

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, 103);