C₂×C₅₃×#1413 #27221109

order	13568
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₅₃×#1413

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#354
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₄⋊Q₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄⋊Q₈×C₅₃ ◁ G

314 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 87 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(54,55);(56,57,59,64,61,66,73,83)(58,67,69,85,68,84,86,81)(60,65,75,82,76,63,72,80)(62,77,78,70,79,71,87,74);(56,58,61,68)(57,62)(59,71,73,77)(60,74,76,70)(63,67)(64,81)(65,84)(66,79)(69,72,86,75)(78,80)(82,87)(83,85)

Conjugacy classes & character table

1484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6585);