C₅₃×#13855 #27221367

order	13568
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅₃×#13855

Subgroups

	subgroup	G/H
center	C₂³×C₄×C₅₃	C₂³
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂⁴	C₂⁴×C₅₃
socle, prime radical	C₂⁴×C₅₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅₃ ◁ G

1246 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 1246 conjugacy classes of subgroups — too many to draw. Only its 494 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 — 4 generators on 81 points:

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

Conjugacy classes & character table

4664 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6843);