C₆₁×#12479 #29515912

order	15616
nilpotency class	4
isoclinism	isoclinic class of #12433
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₆₁×#12479

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₆₁
prime radical	C₆₁×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ C₈×Q₈×C₆₁ ◁ G

216 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(18,22)(19,23)(20,26)(24,28)(25,32)(27,31)(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);(2,6)(7,15)(8,13)(9,16)(17,18,23,30,20,25,27,28,21,24,31,32,26,29,19,22);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(18,24)(22,28)(25,29)(30,32)

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, 5467);