C₅₃×#8623 #27216135

order	13568
nilpotency class	3
isoclinism	isoclinic class of #8608
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#8623

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#380
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

362 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 6784

A minimal faithful permutation representation — 3 generators on 85 points:

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

Conjugacy classes & character table

2756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1611);