#27355866

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1584 ⋊ C₂
Sylow	C₅₃ ⋊ #18068

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1584	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1584	D₅₃
hypocenter	C₅₃	#18068
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1584

586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=fg,e^2=g,f^2=h,b^a=be,e^a=eg,d^b=dh,e^b=eg,d^c=dh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

1058 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 141342);