#27275616

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1532 ⋊ C₂
Sylow	C₅₃ ⋊ #8349

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1532	C₂
socle	C₂²×C₅₃	#379
prime radical	#20288427	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1532	D₅₃
hypocenter	C₅₃	#8349
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1532

720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 points:

(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)(62,65,67,76)(63,69,70,83)(64,73,74,87)(66,77,79,89)(68,80,72,86)(71,82,84,93)(75,88,78,90)(81,91,85,92);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55)(56,60)(57,59)(58,61)(62,64)(63,68)(65,73)(66,78)(67,74)(69,80)(70,72)(71,85)(75,79)(76,87)(77,90)(81,84)(82,92)(83,86)(88,89)(91,93);(55,58,59,60)(62,63,66,71,67,70,79,84)(64,72,75,85,74,68,78,81)(65,69,77,82,76,83,89,93)(73,80,88,91,87,86,90,92)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61092);