#27472002

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1907 ⋊ C₂
Sylow	C₅₃ ⋊ #28137

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1907	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#20283940	C₂
hypercenter	#1907	D₅₃
hypocenter	C₅₃	#28137
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1907

988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 77 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)(63,70)(64,74)(69,71)(73,75);(63,69)(64,73)(66,67)(68,72)(70,71)(74,75);(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)(54,56,57,61)(55,58,59,60)(62,64,66,73,65,74,67,75)(63,68,69,77,70,72,71,76);(54,55)(56,60)(57,59)(58,61)(62,63)(64,72)(65,70)(66,69)(67,71)(68,74)(73,76)(75,77)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 257478);