C₅₃×#13512 #27221024

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#13512

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2085
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#405 ◁ G

232 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 points:

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

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6500);