C₅₃×#13497 #27221009

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13497
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#13497

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2075
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂×#41×C₅₃	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

274 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

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, 6485);