C₆₁×#12282 #29515715

order	15616
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#12282

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂⁴×C₆₁	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₂²⋊C₈×C₆₁ ◁ G

344 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 3904

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

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

Conjugacy classes & character table

2074 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5270);