#27280786

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9294

Subgroups

	subgroup	G/H
center	C₂²	#16810503
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#3208	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂×#127×C₅₃	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3208	D₅₃
hypocenter	C₅₃	#9294
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3208

490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 77 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)(54,55,59,56)(57,61,58,60)(62,65)(63,68)(64,71)(66,75)(67,74)(69,77)(70,73)(72,76);(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)(54,56,59,55)(57,60,58,61)(62,64,70,63,67,72,66,69)(65,71,73,68,74,76,75,77);(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)(54,55,58,61,59,56,57,60)(62,63)(64,70)(65,68,74,77)(66,72)(67,69)(71,75,76,73)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfgh,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,i^a=i^52,d^b=dh,e^b=egh,f^b=fh,i^b=i^52,d^c=dgh,e^c=eh,f^c=fgh

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, 66262);