C₅₃×#7302 #27214814

order	13568
nilpotency class	4
isoclinism	isoclinic class of #7262
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7302

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#340
commutator	C₂×C₈	C₄²×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
prime radical	C₂³×C₅₃	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

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

normal subgroups only

This group has 238 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) = 3392

A minimal faithful permutation representation — 2 generators on 77 points:

(1,2,4,7,5,3,8,6)(10,15,16,19)(11,12)(13,21)(17,24,23,18)(20,22)(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);(1,3)(2,5)(4,6)(7,8)(9,10,12,17,21,24,11,19,14,16,22,23,13,18,20,15)

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, 290);