C₂×C₅₃×#1414 #27221113

order	13568
nilpotency class	5
isoclinism	isoclinic class of #1414
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₅₃×#1414

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#354
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	#124	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#124×C₅₃ ◁ G

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

normal subgroups only

This group has 340 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)(58,66,68,81)(59,70,72,86)(60,74,76,64)(61,65)(62,67,77,84)(63,73)(69,83,75,82)(71,78,85,79)(80,87);(56,58)(57,62,65,79)(59,71)(60,75)(61,69)(63,66,80,83)(64,77,74,78)(67,72)(68,76)(70,81,86,82)(73,85)(84,87)

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, 6589);