#27530938

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2081 ⋊ C₂
Sylow	C₅₃ ⋊ #32698

Subgroups

	subgroup	G/H
center	C₂	#20281943
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2081	C₂
socle	C₂×C₅₃	#2749
prime radical	#20275731	C₂
hypercenter	#2081	D₅₃
hypocenter	C₅₃	#32698
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2081

594 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)(56,64)(59,67)(62,69)(66,68);(54,57)(55,61,63,65)(56,62,64,69)(58,60)(59,66)(67,68);(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)(55,60)(57,63)(58,64)(59,65)(61,67)(62,66)(68,69);(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)(54,55,59,62,58,63,67,69)(56,60,61,68,64,57,65,66)

Polycyclic presentation:

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

Conjugacy classes & character table

632 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 316414);