#27390055

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #21477

Subgroups

	subgroup	G/H
center	C₄	#120×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3018	C₂
socle	C₂×C₅₃	C₄×#120
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#3018	D₅₃
hypocenter	C₅₃	#21477
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3018

1290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 175531);