#27352226

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1532 ⋊ C₂
Sylow	C₅₃ ⋊ #17678

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1532	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	#20288009	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1532	D₅₃
hypocenter	C₅₃	#17678
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1532

976 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)(54,57,59,58)(55,61,62,66)(56,64,65,67)(60,68,63,69)(71,78)(76,85)(77,83)(79,84);(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)(70,73,75,74)(71,77,78,83)(72,80,82,81)(76,84,85,79);(54,56)(55,60)(57,64)(58,67)(59,65)(61,68)(62,63)(66,69)(70,72)(71,76)(73,80)(74,81)(75,82)(77,84)(78,85)(79,83);(54,55,58,61,59,62,57,66)(56,63,64,68,65,60,67,69)(70,71,74,77,75,78,73,83)(72,79,81,76,82,84,80,85)

Polycyclic presentation:

2^8,53|a^2=f,c^2=h,d^2=gh,e^2=h,f^2=g,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh,i^c=i^52

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 137702);