C₅₃×#10319 #27217831

order	13568
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#10319

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

538 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 13568

A minimal faithful permutation representation — 3 generators on 89 points:

(1,2)(3,4)(5,6,10,14)(7,15,18,19)(8,13,9,12)(11,17,20,16)(21,22,26,30)(23,31,34,35)(24,29,25,28)(27,32,36,33)(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);(2,4)(5,7,10,18)(6,11,14,20)(8,17,9,16)(12,19,13,15)(23,32)(24,25)(27,35)(28,29)(31,36)(33,34);(6,12)(8,9)(11,19)(13,14)(15,20)(16,17)(21,23,26,34)(22,27,30,36)(24,32,25,33)(28,35,29,31)

Conjugacy classes & character table

2756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 3307);