C₆₁×#7389 #29510822

order	15616
nilpotency class	5
isoclinism	isoclinic class of #7389
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₆₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₆₁×#7389

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	#1333
commutator	C₁₆	C₂×C₈×C₆₁
Frattini, cosocle	#346	C₂²×C₆₁
prime radical	C₆₁×#508	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×C₄×C₆₁ ◁ C₄×C₈×C₆₁ ◁ C₆₁×#346 ◁ G

118 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 118 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

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, 377);