#27422167

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1793 ⋊ C₂
Sylow	C₅₃ ⋊ #24458

Subgroups

	subgroup	G/H
center	C₂²	#16811255
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1793	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#16811431	C₂
hypercenter	#1793	D₅₃
hypocenter	C₅₃	#24458
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1793

1328 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)(59,66)(60,70)(62,75)(63,76)(64,73)(65,79)(67,83)(68,82)(69,84)(71,87)(72,86)(74,77)(78,85)(80,88)(81,89);(55,57)(58,60,63,72)(59,65,68,81)(61,70,76,86)(62,71,77,69)(64,78,67,80)(66,79,82,89)(73,85,83,88)(74,84,75,87);(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)(59,64)(60,69)(62,77)(66,73)(67,68)(70,84)(71,72)(74,75)(78,80)(82,83)(85,88)(86,87);(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)(54,55,56,57)(58,59)(60,65)(61,73)(62,64)(63,68)(66,74)(67,77)(69,80)(70,85)(71,78)(72,81)(75,82)(76,83)(79,84)(86,88)(87,89)

Polycyclic presentation:

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

Conjugacy classes & character table

875 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 207643);