C₆₁×#7319 #29510752

order	15616
nilpotency class	4
isoclinism	isoclinic class of #7318
automorphism	?
homology	Abelianization (H₁) = C₄²×C₆₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₆₁×#7319

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#1304
commutator	C₂²×C₄	C₄²×C₆₁
Frattini, cosocle	C₂²×#41	C₂²×C₆₁
prime radical	C₂²×#41×C₆₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂³×C₆₁ ◁ C₂²×#41×C₆₁ ◁ G

456 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

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, 307);