#27356368

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1604 ⋊ C₂
Sylow	C₅₃ ⋊ #18123

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1604	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288043	C₂
hypercenter	#1604	D₅₃
hypocenter	C₅₃	#18123
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1604

1208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

644 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 141844);