#27404944

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #22896

Subgroups

	subgroup	G/H
center	C₂²	#16811255
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#455	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×H(5,𝔽₂)×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#455	D₅₃
hypocenter	C₅₃	#22896
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#455

1016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 points:

(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)(58,61)(59,67)(60,73)(62,79)(63,76)(64,77)(65,81)(66,68)(69,83)(70,85)(71,87)(72,78)(74,89)(75,88)(80,82)(84,86);(58,61,64,77)(59,66,70,83)(60,72,75,89)(62,76,63,79)(65,80,82,81)(67,69,85,68)(71,87,86,84)(73,74,88,78);(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)(54,56,57,55)(58,60)(59,65)(61,72)(62,78)(63,74)(64,75)(66,80)(67,84)(68,86)(69,71)(70,82)(73,76)(77,89)(79,88)(81,83)(85,87);(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)(54,55,57,56)(58,59)(60,71)(61,67)(62,69)(63,68)(64,70)(65,74)(66,76)(72,80)(73,87)(75,86)(77,85)(78,82)(79,83)(81,89)(84,88)

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 190420);