#27365047

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3253 ⋊ C₂
Sylow	C₅₃ ⋊ #18965

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3253	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#3253	D₅₃
hypocenter	C₅₃	#18965
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3253

900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150523);