C₆₁×#12300 #29515733

order	15616
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#12300

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₆₁
prime radical	C₆₁×#3197	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₂×M₁₆×C₆₁ ◁ G

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

normal subgroups only

This group has 328 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 — 3 generators on 93 points:

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

Conjugacy classes & character table

1525 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5288);