C₆₁×#29657 #29533090

order	15616
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#29657

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₆₁
prime radical	C₂⁵×C₆₁	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂⁵×C₆₁ ◁ G

1148 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 242 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 — 4 generators on 93 points:

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

Conjugacy classes & character table

2257 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 22645);