#27356299

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1599 ⋊ C₂
Sylow	C₅₃ ⋊ #18114

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1599	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#1599	D₅₃
hypocenter	C₅₃	#18114
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1599

1246 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)(62,65,67,66)(63,69,70,74)(64,72,73,75)(68,76,77,71);(54,56,57,60)(55,58,59,61);(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)(54,55)(56,58)(57,59)(60,61)(62,64)(63,68)(65,72)(66,75)(67,73)(69,76)(70,77)(71,74);(58,61)(62,63)(64,71)(65,69)(66,74)(67,70)(68,72)(73,76)(75,77)

Polycyclic presentation:

2^8,53|c^2=h,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,e^b=eg,f^b=fh,i^b=i^52,e^c=eh,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, 141775);