C₅₃×#7490 #27215002

order	13568
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#7490

Subgroups

	subgroup	G/H
center	C₄×C₅₃	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₄²×C₅₃
Frattini, cosocle	#532	C₂²×C₅₃
socle	C₂×C₅₃	#1390
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#532 ◁ G

274 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 2 generators on 85 points:

(1,2)(3,13,15,30)(4,18)(5,10)(6,11,22,29)(7,12)(8,14,23,31)(9,20,28,21)(16,24,17,27)(19,25,32,26)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,5,14,7,17,4,19)(2,8,10,24,12,26,18,30)(6,15,21,31,22,16,20,32)(9,27,29,25,28,13,11,23)

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, 478);