C₅₃×#7312 #27214824

order	13568
nilpotency class	4
isoclinism	isoclinic class of #7310
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#7312

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1304
commutator	C₂⁴	C₄²×C₅₃
Frattini, cosocle	C₂²×#41	C₂²×C₅₃
prime radical	C₂²×#41×C₅₃	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂³×C₅₃ ◁ C₂²×#41×C₅₃ ◁ G

456 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 74 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:

(1,2,4,9)(3,7,11,20)(5,17,18,24)(6,8,15,23)(10,16,25,19)(12,27,28,31)(13,22,14,30)(21,26,32,29)(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);(1,3,5,12,6,13,19,29)(2,7,10,21,8,22,17,27)(4,14,16,28,15,11,18,26)(9,20,24,31,23,30,25,32)

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, 300);