C₅₃×#9008 #27216520

order	13568
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅₃×#9008

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅₃	#120
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#386
prime radical	C₅₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

398 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 101
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 101 points:

(1,2,6,9,5,10,4,8)(3,7,13,15,12,16,11,14)(17,18,22,27,23,28,21,26)(19,24,30,38,31,39,36,42)(20,32,35,25,33,41,34,40)(29,43,45,48,44,47,46,37)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(1,2,5,10)(3,7,12,16)(4,9,6,8)(11,15,13,14)(17,19,22,30,23,31,21,36)(18,24,27,38,28,39,26,42)(20,29,34,46,33,44,35,45)(25,37,32,47,40,48,41,43);(1,3,5,12)(2,7,10,16)(4,13,6,11)(8,15,9,14)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,37)(26,41)(27,32)(28,40)(30,46)(31,44)(36,45)(38,47)(39,48)(42,43)

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1996);