C₆₁×#7129 #29510562

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

Aliases

direct product

C₆₁×#7129

Subgroups

	subgroup	G/H
center	C₄×C₆₁	#329
commutator	C₄²	C₂×C₈×C₆₁
Frattini, cosocle	#376	C₂²×C₆₁
socle	C₂×C₆₁	#1272
prime radical	C₂³×C₆₁	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#376 ◁ G

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

normal subgroups only

This group has 210 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)(3,11,13,26,14,27,19,30)(4,15,17,31)(5,9)(6,10)(7,12,21,28,23,16,22,29)(8,18,24,32)(20,25)(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);(1,3,6,14)(2,7)(4,16,17,12)(5,19,20,13)(8,11)(9,21)(10,23)(15,30)(18,29,32,28)(22,25)(24,27)(26,31)

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, 117);