#27351671

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1515 ⋊ C₂
Sylow	C₅₃ ⋊ #17601

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1515	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	#13133191	C₂³
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1515	D₅₃
hypocenter	C₅₃	#17601
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1515

882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 85 points:

(1,2)(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)(70,73,75,81)(71,77,78,85)(72,79,74,80)(76,83,82,84);(55,60)(59,67)(61,68)(62,69)(70,72,75,74)(71,76,78,82)(73,80,81,79)(77,84,85,83);(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)(54,56)(55,59)(57,65)(58,64)(60,67)(61,69)(62,68)(63,66)(70,71)(72,76)(73,77)(74,82)(75,78)(79,83)(80,84)(81,85);(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)(56,62,64,69)(57,61)(58,60)(59,63,67,65)(66,68)(71,76)(72,74)(77,83)(78,82)(79,80)(84,85)

Polycyclic presentation:

2^8,53|a^2=f,c^2=g,d^2=g,e^2=h,b^a=be,c^a=ch,e^a=eg,i^a=i^52,e^b=eh,f^b=fgh,i^b=i^52,d^c=dg,i^d=i^52

Conjugacy classes & character table

725 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 137147);