#27416594

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2031 ⋊ C₂
Sylow	C₅₃ ⋊ #23990

Subgroups

	subgroup	G/H
center	C₂	#20278035
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2031	C₂
socle	C₂×C₅₃	#2404
hypercenter	#2031	D₅₃
hypocenter	C₅₃	#23990
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2031

1260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 4 generators on 69 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)(54,56)(55,60)(57,63)(58,62)(59,66)(61,67)(64,68)(65,69);(54,55)(56,60)(57,65)(58,61)(59,64)(62,67)(63,69)(66,68);(55,59,61,65)(60,66,67,69);(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)(57,64)(59,65)(60,67)(63,68)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 202070);