#27596503

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₄×#123×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #36916

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₄×#123×C₅₃	C₂
prime radical	#20281080	C₂
hypercenter	C₄×#123	D₅₃
hypocenter	C₅₃	#36916
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄×#123

1650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 85 points:

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=f,c^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,i^a=i^52,c^b=ch,d^b=dfh

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 381979);