#27473109

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1849 ⋊ C₂
Sylow	C₅₃ ⋊ #28244

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133192
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1849	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811173	C₂²
hypercenter	#1849	D₅₃
hypocenter	C₅₃	#28244
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1849

838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 89 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)(58,62,64,74,65,73,63,60)(59,68,70,66,71,83,69,80)(61,77,81,76,79,88,78,72)(67,85,87,82,86,89,75,84);(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)(55,57)(58,61)(59,67)(60,72)(62,77)(63,78)(64,81)(65,79)(66,82)(68,85)(69,75)(70,87)(71,86)(73,88)(74,76)(80,84)(83,89);(54,55,56,57)(58,60,64,62,65,74,63,73)(59,66,70,83,71,80,69,68)(61,76,81,88,79,72,78,77)(67,84,87,85,86,82,75,89);(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)(58,59,65,71)(60,68,74,83)(61,75,79,87)(62,80,73,66)(63,70,64,69)(67,81,86,78)(72,84,76,82)(77,89,88,85)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 258585);