C₅₃×#11390 #27218902

order	13568
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅₃×#11390

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator	C₂⁴	C₂²×C₄×C₅₃
Frattini, cosocle	C₂⁵	C₂³×C₅₃
prime radical	C₂⁵×C₅₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂⁵×C₅₃ ◁ G

1358 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 1358 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 73 points:

(1,2,3,4)(13,14)(15,19,20,17)(16,18)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(5,6,8,10)(7,11,12,9)(13,15)(14,17)(16,20)(18,19);(5,7,8,12)(6,9,10,11)(13,14)(15,17)(16,18)(19,20)

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, 4378);