#27530912

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #32697

Subgroups

	subgroup	G/H
center	C₂	#20281932
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1915	C₂
socle	C₂×C₅₃	#2749
prime radical	C₂×#41×C₅₃	C₂³
hypercenter	#1915	D₅₃
hypocenter	C₅₃	#32697
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1915

528 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)(54,58,59,75)(55,63,65,71)(56,67,68,74)(57,61,72,80)(60,79,77,76)(62,78,70,69)(64,84,85,66)(73,83,82,81);(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)(54,57,59,72)(55,62)(56,66)(58,61,75,80)(60,73,77,82)(63,78)(64,67)(65,70)(68,84)(69,71)(74,85)(76,81,79,83);(54,56)(55,61)(57,71)(58,74)(59,68)(60,78)(62,79)(63,72)(64,83)(65,80)(66,73)(67,75)(69,77)(70,76)(81,85)(82,84);(54,55,60,64,59,65,77,85)(56,61,69,81,68,80,78,83)(57,70,73,74,72,62,82,67)(58,63,76,66,75,71,79,84)

Polycyclic presentation:

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

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, 316388);