C₅₃×#7808 #27215320

order	13568
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7808

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

404 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 77 points:

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

Conjugacy classes & character table

1961 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 796);