#27365288

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3171 ⋊ C₂
Sylow	C₅₃ ⋊ #18990

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133185
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3171	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#3171	D₅₃
hypocenter	C₅₃	#18990
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3171

862 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)(70,72,74,79)(71,76,77,83)(73,78,80,84)(75,82,81,85);(54,56)(55,59)(57,66)(58,63)(60,69)(61,68)(62,64)(65,67)(73,80)(75,81)(78,84)(82,85);(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)(54,55,58,61)(56,59,63,68)(57,65,64,60)(62,69,66,67)(70,71,74,77)(72,76,79,83)(73,81,80,75)(78,85,84,82);(56,62)(57,64)(59,67)(60,65)(63,66)(68,69)(71,75)(73,80)(76,82)(77,81)(78,84)(83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150764);