#27277287

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₄×#108×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #8672

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#108×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂×#11719945	Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#108	D₅₃
hypocenter	C₅₃	#8672
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 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)(54,56)(55,58)(57,59)(60,61)(64,72)(68,76)(71,77)(73,74);(62,64,65,73,67,72,66,74)(63,68,69,71,70,76,75,77);(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)(54,55,57,60,58,56,61,59)(62,63,66,69,67,70,65,75)(64,71,74,76,72,77,73,68)

Polycyclic presentation:

2^8,53|a^2=f,b^2=d,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,i^a=i^52,c^b=cg,e^c=eh,i^c=i^52

Conjugacy classes & character table

1502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 62763);