#27280590

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9266

Subgroups

	subgroup	G/H
center	C₂²	#16810503
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#3232	C₂
socle	C₂²×C₅₃	#386
prime radical	#121×C₅₃	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3232	D₅₃
hypocenter	C₅₃	#9266
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3232

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(54,56)(55,61)(57,65)(58,67)(59,66)(60,68)(62,64)(63,69)(70,72)(71,76)(73,81)(74,79)(75,83)(77,85)(78,82)(80,84);(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)(55,60,63,62)(61,68,69,64)(70,71,73,80,74,77,82,75)(72,76,81,84,79,85,78,83);(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)(54,55,57,60,59,63,58,62)(56,64,65,61,66,68,67,69)(71,75,77,80)(72,78,79,81)

Polycyclic presentation:

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

Conjugacy classes & character table

875 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 66066);