#27415535

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1905 ⋊ C₂
Sylow	C₅₃ ⋊ #23885

Subgroups

	subgroup	G/H
center	C₂	#20278027
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1905	C₂
socle	C₂×C₅₃	#2403
hypercenter	#1905	D₅₃
hypocenter	C₅₃	#23885
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1905

1662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 201011);