C₂²×C₅₃×#458 #27267905

order	13568
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×C₅₃×#458

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂²×H(5,𝔽₂)×C₅₃	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂⁵×C₅₃ ◁ G

5294 conjugacy classes of subgroups; only the 1134 normal subgroups are shown.

normal subgroups only

This group has 5294 conjugacy classes of subgroups — too many to draw. Only its 1134 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 — 5 generators on 73 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);(58,59,62,65)(60,63,67,70)(61,64,68,71)(66,72,73,69);(58,60)(59,63)(61,66)(62,67)(64,69)(65,70)(68,73)(71,72);(58,61)(59,64)(60,66)(62,68)(63,69)(65,71)(67,73)(70,72)

Conjugacy classes & character table

3392 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 53381);