C₆₁×#23805 #29527238

order	15616
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₆₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₆₁×#23805

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄³×C₆₁ ◁ G

974 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 286 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) = 89
transitive degree	μ_t(G) = 15616

A minimal faithful permutation representation — 4 generators on 89 points:

(5,6,8,9)(7,10,11,12)(13,14,17,22,19,21,18,23)(15,24,25,28,16,27,26,20)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,2)(3,4)(6,9)(10,12)(13,15,19,16)(14,20,21,28)(17,26,18,25)(22,27,23,24);(2,4)(13,16,19,15)(14,20,21,28)(17,25,18,26)(22,27,23,24);(5,7,8,11)(6,10,9,12)(14,21)(20,28)(22,23)(24,27)

Conjugacy classes & character table

3172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 16793);