C₅₃×#13564 #27221076

order	13568
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#13564

Subgroups

	subgroup	G/H
center	C₄×C₅₃	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₅₃
Frattini, cosocle	#124	C₂³×C₅₃
socle	C₂×C₅₃	C₂×AΣL(1,𝔽₁₆)
prime radical	C₅₃×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#532 ◁ G

494 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 69 points:

(1,2,5,9,6,3,11,13)(4,7,14,12,15,10,16,8)(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)(2,6)(4,10)(5,12)(7,15)(8,11)(9,16)(13,14);(1,4,6,15)(2,7,3,10)(5,14,11,16)(8,9,12,13)

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6552);