C₅₃×#13074 #27220586

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13074
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅₃×#13074

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1893
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₅₃×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

490 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 1696

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

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

Conjugacy classes & character table

2438 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6062);