C₅₃×#7327 #27214839

order	13568
nilpotency class	4
isoclinism	isoclinic class of #7307
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅₃×#7327

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#342
commutator	C₂²×C₄	C₄²×C₅₃
Frattini, cosocle	C₂×#117	C₂²×C₅₃
prime radical	C₂⁴×C₅₃	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 101
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 101 points:

(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,21,25,22,26,34,41)(19,27,29,42,30,43,33,47)(20,24,31,38,32,39,46,48)(23,35,36,28,37,44,40,45)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,23)(20,28,32,45)(21,33)(22,30)(24,27,39,43)(25,40)(26,37)(29,34)(31,35,46,44)(36,41)(38,47,48,42)

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, 315);