C₅₃×#13514 #27221026

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

Aliases

direct product

C₅₃×#13514

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

298 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 130 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)(7,12)(8,10)(9,11)(13,14,15,19,23,27,28,26,17,21,24,25,16,20,22,18)(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)(5,7,9,8)(6,10,11,12);(2,4)(5,8,9,7)(6,10,11,12)(14,18)(15,22)(16,23)(19,20)(21,26)(24,28)(25,27)

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, 6502);