C₆₁×#13611 #29517044

order	15616
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#13611

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#2127
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₄⋊Q₈	C₂³×C₆₁
prime radical	C₆₁×#3221	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₆₁×#536 ◁ G

316 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

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, 6599);