#27419967

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1979 ⋊ C₂
Sylow	C₅₃ ⋊ #24282

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1979	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811135	C₂²
hypercenter	#1979	D₅₃
hypocenter	C₅₃	#24282
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1979

1308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 205443);