C₅₃×#13486 #27220998

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13486

Subgroups

	subgroup	G/H
center	C₄×C₅₃	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#2071
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

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

normal subgroups only

This group has 296 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:

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

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, 6474);