#27416554

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1889 ⋊ C₂
Sylow	C₅₃ ⋊ #23986

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#1889	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20275838	C₂
hypercenter	#1889	D₅₃
hypocenter	C₅₃	#23986
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #1889

1426 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)(62,65,66,72)(63,68,69,75)(64,70,71,76)(67,74,73,77);(67,73)(74,77);(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)(63,67)(65,70)(66,71)(68,74)(69,73)(72,76)(75,77);(54,55)(56,60)(57,59)(58,61)(62,63)(64,67)(65,68)(66,69)(70,74)(71,73)(72,75)(76,77)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 202030);