C₆₁×#60400 #29563833

order	15616
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₆₁ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₆₁×#60400

Subgroups

	subgroup	G/H
center, socle	C₂×C₆₁	C₂²×#120
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₁
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂²×Q₈×C₆₁ ◁ G

4732 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.

normal subgroups only

This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 1114 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 — 5 generators on 93 points:

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

Conjugacy classes & character table

3538 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 53388);