C₅₃×#7191 #27214703

order	13568
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₅₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₃×#7191

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#99
commutator	C₂³	C₄×C₈×C₅₃
Frattini, cosocle	C₂²×M₁₆	C₂²×C₅₃
socle	C₂²×C₅₃	#323
prime radical	C₂⁴×C₅₃	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂²×M₁₆×C₅₃ ◁ G

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

normal subgroups only

This group has 342 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) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 73 points:

(5,6,7,11,8,12,14,18,9,10,13,16,15,17,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(1,2,3,4)(6,10)(12,17)

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, 179);