C₅₃×#13562 #27221074

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13561
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅₃×#13562

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₅₃
Frattini, cosocle	#127	C₂³×C₅₃
prime radical	C₅₃×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#532 ◁ G

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

normal subgroups only

This group has 490 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:

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

Conjugacy classes & character table

1484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6550);