C₅₃×#7230 #27214742

order	13568
nilpotency class	4
isoclinism	isoclinic class of #7217
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#7230

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₃	#340
commutator	C₂×C₈	C₄²×C₅₃
Frattini, cosocle	C₂³×C₈	C₂²×C₅₃
prime radical	C₂⁴×C₅₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

314 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 110 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)(3,7,8,5)(4,6)(10,15)(11,19)(12,21)(13,20)(16,17)(18,24)(22,23)(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)(4,8)(6,7)(9,10,12,16,13,17,11,15,14,18,19,23,20,22,21,24)

Conjugacy classes & character table

2756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 218);