C₂²×C₅₃×B(4,𝔽₂) #27267904

order	13568
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅₃ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×C₅₃×B(4,𝔽₂)

Subgroups

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

7014 conjugacy classes of subgroups; only the 1134 normal subgroups are shown.

normal subgroups only

This group has 7014 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) = 65
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 5 generators on 65 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);(62,65);(58,59)(60,62)(61,63)(64,65);(58,60)(59,62)(61,64)(63,65)

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, 53380);