#27358414

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3236 ⋊ C₂
Sylow	C₅₃ ⋊ #18341

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3236	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288107	C₂
hypercenter	#3236	D₅₃
hypocenter	C₅₃	#18341
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3236

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 143890);