#27423051

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2064 ⋊ C₂
Sylow	C₅₃ ⋊ #24525

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2064	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×#11719949	C₂²
hypercenter	#2064	D₅₃
hypocenter	C₅₃	#24525
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2064

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(55,59)(58,60)(62,66,68,67)(63,71,73,72)(64,75,77,76)(65,80,81,82)(69,84,86,85)(70,78,88,89)(74,91,92,79)(83,90,93,87);(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)(62,65)(63,70)(64,74)(66,80)(67,82)(68,81)(69,83)(71,78)(72,89)(73,88)(75,91)(76,79)(77,92)(84,90)(85,87)(86,93);(54,56,57,61)(55,58,59,60)(62,64)(63,69)(65,79)(66,75)(67,76)(68,77)(70,87)(71,84)(72,85)(73,86)(74,80)(78,83)(81,91)(82,92)(88,90)(89,93);(54,55)(56,60)(57,59)(58,61)(62,63,66,71,68,73,67,72)(64,69,75,84,77,86,76,85)(65,78,82,70,81,89,80,88)(74,90,79,83,92,87,91,93)

Polycyclic presentation:

2^8,53|a^2=fh,b^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 208527);