C₆₁×#11436 #29514869

order	15616
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#11436

Subgroups

	subgroup	G/H
center	C₄×C₆₁	C₂×#99
commutator	C₂³	C₂²×C₈×C₆₁
Frattini, cosocle	C₂×M₁₆	C₂³×C₆₁
socle	C₂×C₆₁	C₂×#323
prime radical	C₂²×#41×C₆₁	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×M₁₆×C₆₁ ◁ G

504 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 93 points:

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

Conjugacy classes & character table

2806 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 4424);