#27374623

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1719 ⋊ C₂
Sylow	C₅₃ ⋊ #19834

Subgroups

	subgroup	G/H
center	C₂²	#16811258
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1719	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#16810815	C₂²
hypercenter	#1719	D₅₃
hypocenter	C₅₃	#19834
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1719

590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 85 points:

(1,2)(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)(70,74,76,71)(72,79,81,85)(73,82,75,78)(77,84,83,80);(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)(54,57,59,65)(55,61,62,69)(56,63,58,67)(60,68,66,64)(70,73,76,75)(71,78,74,82)(72,80,81,84)(77,85,83,79);(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)(54,56,59,58)(55,60,62,66)(57,63,65,67)(61,68,69,64)(70,72,76,81)(71,77,74,83)(73,80,75,84)(78,85,82,79);(54,55)(56,60)(57,64)(58,66)(59,62)(61,63)(65,68)(67,69)(70,71,76,74)(72,79,81,85)(73,78,75,82)(77,84,83,80)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 160099);