#27484042

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2031 ⋊ C₂
Sylow	C₅₃ ⋊ #29154

Subgroups

	subgroup	G/H
center	C₄	#16811335
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2031	C₂
socle	C₂×C₅₃	#2563
prime radical	C₂×#13133237	C₂²
hypercenter	#2031	D₅₃
hypocenter	C₅₃	#29154
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2031

612 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 269518);