C₂×C₅₃×#1711 #27227111

order	13568
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×C₅₃×#1711

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

802 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 286 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) = 6784

A minimal faithful permutation representation — 4 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)(58,64)(59,68)(60,62)(61,66)(63,65)(67,70)(69,71)(72,73)(74,80)(75,77)(76,84)(78,82)(79,83)(81,86)(85,87);(56,58,65,71,60,67,59,66)(57,61,68,70,62,69,63,64)(73,77)(78,85)(79,84)(80,86);(57,62)(61,69)(63,68)(64,70)(72,74,76,82,75,81,83,87)(73,78,79,80,77,85,84,86)

Conjugacy classes & character table

4028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 12587);