C₆₁×#10143 #29513576

order	15616
nilpotency class	3
isoclinism	isoclinic class of #10128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#10143

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂³×C₆₁	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

434 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 7808

A minimal faithful permutation representation — 3 generators on 93 points:

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

Conjugacy classes & character table

1891 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 3131);