#27383369

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1791 ⋊ C₂
Sylow	C₅₃ ⋊ #20612

Subgroups

	subgroup	G/H
center	C₄	#120×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1791	C₂
socle	C₂×C₅₃	C₂×#379
prime radical	#585×D₅₃	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1791	D₅₃
hypocenter	C₅₃	#20612
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1791

3640 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)(55,62)(60,64)(61,74)(63,76)(72,78)(73,77)(75,84)(83,85);(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)(54,57)(55,61)(56,65)(58,69)(59,68)(60,72)(62,74)(63,75)(64,78)(66,79)(67,80)(70,82)(71,81)(73,83)(76,84)(77,85);(54,56)(55,60)(57,65)(58,70)(59,66)(61,72)(62,64)(63,77)(67,71)(68,79)(69,82)(73,76)(74,78)(75,85)(80,81)(83,84);(54,55,58,63,59,62,71,76)(56,64,67,77,66,60,70,73)(57,61,69,75,68,74,81,84)(65,72,80,83,79,78,82,85)

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 168845);