C₅₃×#11670 #27219182

order	13568
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#11670

Subgroups

	subgroup	G/H
center	C₄×C₅₃	#418
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1490
prime radical	C₅₃×#575	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₄×M₁₆×C₅₃ ◁ G

348 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Conjugacy classes & character table

2438 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 4658);