C₅₃×#32704 #27240216

order	13568
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅₃×#32704

Subgroups

	subgroup	G/H
center	C₄×C₅₃	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂×C₅₃	#2804
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ G

556 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 4 generators on 69 points:

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

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 25692);