C₅₃×#13079 #27220591

order	13568
nilpotency class	4
isoclinism	isoclinic class of #13077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅₃×#13079

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1893
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₅₃×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

486 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 85 points:

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

Conjugacy classes & character table

2438 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6067);