#27275754

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1786 ⋊ C₂
Sylow	C₅₃ ⋊ #8375

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1786	C₂
socle	C₂²×C₅₃	#379
prime radical	C₂×D₄×C₂²⋊D₅₃	C₂
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1786	D₅₃
hypocenter	C₅₃	#8375
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1786

1518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 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)(55,58)(57,61)(59,60);(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)(55,58)(57,61)(59,60)(63,67)(64,65)(68,69);(54,55)(56,60,61,58)(57,59)(62,63,65,67,66,69,64,68)

Polycyclic presentation:

2^8,53|a^2=f,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,f^b=fgh,i^b=i^52,f^c=fg,i^c=i^52

Conjugacy classes & character table

1031 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61230);