C₅₃×#7194 #27214706

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

Aliases

direct product

C₅₃×#7194

Subgroups

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

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

normal subgroups only

This group has 310 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) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 89 points:

(5,6,9,14,10,13,24,29,11,15,25,31,8,21,22,30)(7,16,18,32,19,28,34,36,20,33,23,35,17,12,26,27)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,2,3,4)(5,7,8,17,11,20,10,19)(6,12,13,16,15,28,21,33)(9,23,22,34,25,18,24,26)(14,27,29,32,31,36,30,35)

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, 182);