C₂×C₅₃×#1407 #27221088

order	13568
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×C₅₃×#1407

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#353
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

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

normal subgroups only

This group has 416 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)(59,70,71,86)(60,73,75,64)(61,65)(62,67,76,83)(63,72)(69,79,74,82)(77,84,78,85)(80,87);(56,58,61,69)(57,62,65,78)(59,67,71,85)(60,74,75,68)(63,79,80,66)(64,76,73,77)(70,82,86,81)(72,83,87,84)

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, 6564);