C₅₃×#7283 #27214795

order	13568
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅₃×#7283

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#99
commutator	C₂²×C₄	C₄²×C₅₃
Frattini, cosocle	C₂×C₄×D₄	C₂²×C₅₃
socle	C₂²×C₅₃	#342
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×D₄×C₅₃ ◁ G

402 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 3392

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

(1,2,4,7,5,8,3,6)(10,13)(11,15)(14,16)(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);(2,6,8,7)(9,10)(11,13,15,16)(12,14)

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, 271);