#27512529

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1904 ⋊ C₂
Sylow	C₅₃ ⋊ #31418

Subgroups

	subgroup	G/H
center	C₂	#20281155
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1904	C₂
socle	C₂×C₅₃	#2688
hypercenter	#1904	D₅₃
hypocenter	C₅₃	#31418
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1904

686 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)(55,62)(59,64)(60,75)(61,79)(63,71)(66,72)(68,80)(70,73)(74,81)(76,77)(78,85)(82,84);(54,57,58,69)(55,61,63,80)(56,65)(59,73,64,70)(60,74)(62,79,71,68)(66,84)(67,83)(72,82)(75,81)(76,85)(77,78);(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)(54,56,64,66,58,67,59,72)(55,60,71,76,63,77,62,75)(57,65,73,82,69,83,70,84)(61,78,79,85,80,74,68,81);(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)(56,60)(57,68)(58,63)(59,71)(61,70)(62,64)(65,81)(66,75)(67,77)(69,79)(72,76)(73,80)(74,82)(78,84)(83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

788 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 298005);