C₆₁×#7483 #29510916

order	15616
nilpotency class	5
isoclinism	isoclinic class of #7481
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#7483

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#1390
commutator	C₂×Q₈	C₄²×C₆₁
Frattini, cosocle	C₂×C₄⋊Q₈	C₂²×C₆₁
prime radical	C₂³×C₆₁	#99
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄⋊Q₈×C₆₁ ◁ G

262 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 93 points:

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

Conjugacy classes & character table

1708 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 471);