C₅₃×#7252 #27214764

order	13568
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₃×#7252

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#99
commutator	C₂²×C₄	C₄²×C₅₃
Frattini, cosocle	C₂×#117	C₂²×C₅₃
socle	C₂²×C₅₃	#342
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#117×C₅₃ ◁ G

344 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 2 generators on 93 points:

(1,2)(3,5)(4,7)(6,8)(9,10,12,16,14,18,24,30)(11,19,21,32,20,15,26,27)(13,17,23,29,25,31,36,39)(22,28,35,38,34,33,37,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(2,5,7,6)(9,11,13,22)(10,15,17,28)(12,21,23,35)(14,20,25,34)(16,27,29,38)(18,19,31,33)(24,26,36,37)(30,32,39,40)

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, 240);