C₆₁×#29658 #29533091

order	15616
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₁ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₆₁×#29658

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 ◁ C₂³×C₄×C₆₁ ◁ G

1086 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 238 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)(4,8,15,20)(6,10,17,22)(12,24,25,19)(14,23,27,18)(16,21,28,31)(26,30,32,29)(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)(4,12,15,25)(6,14,17,27)(8,19,20,24)(10,18,22,23)(16,26,28,32)(21,29,31,30);(4,15)(7,18)(8,20)(11,23)(12,25)(16,28)(19,30)(21,31)(24,29)(26,32);(1,4,6,16)(2,8,10,21)(3,12,14,26)(5,15,17,28)(7,19,18,29)(9,20,22,31)(11,24,23,30)(13,25,27,32)

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, 22646);