C₅₃×#7357 #27214869

order	13568
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₅₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅₃×#7357

Subgroups

	subgroup	G/H
center	C₄×C₅₃	#353
commutator	C₄²	C₂×C₈×C₅₃
Frattini, cosocle	#423	C₂²×C₅₃
socle	C₂×C₅₃	#1318
prime radical	C₅₃×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#423 ◁ G

222 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Conjugacy classes & character table

1802 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 345);