C₅₃×#8741 #27216253

order	13568
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#8741

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#381
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

418 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 3392

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

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

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1729);