C₅₃×#7310 #27214822

order	13568
nilpotency class	4
isoclinism	isoclinic class of #7310
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅₃×#7310

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#1304
commutator	C₂⁴	C₄²×C₅₃
Frattini, cosocle	C₂³×D₄	C₂²×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂³×C₅₃ ◁ C₂³×D₄×C₅₃ ◁ G

650 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 2 generators on 69 points:

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

Conjugacy classes & character table

1484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 298);