#27351887

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1509 ⋊ C₂
Sylow	C₅₃ ⋊ #17624

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1509	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	#16811168	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1509	D₅₃
hypocenter	C₅₃	#17624
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1509

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 85 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)(54,57,59,58)(55,62,63,61)(56,65,66,67)(60,68,69,64)(72,78)(76,83)(79,84)(82,85);(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)(54,57,59,58)(55,61,63,62)(56,65,66,67)(60,64,69,68)(70,72)(71,76)(73,79)(74,78)(75,82)(77,83)(80,85)(81,84);(54,56)(55,60)(57,65)(58,67)(59,66)(61,64)(62,68)(63,69)(70,71)(72,76)(73,80)(74,77)(75,81)(78,83)(79,85)(82,84);(54,55)(56,64)(57,62)(58,61)(59,63)(60,65)(66,68)(67,69)(71,75,77,80)(76,82,83,85)

Polycyclic presentation:

2^8,53|a^2=f,c^2=g,d^2=g,e^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgh,i^d=i^52

Conjugacy classes & character table

881 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 137363);