C₅₃×#13530 #27221042

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13530

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

358 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 81
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 81 points:

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

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, 6518);