#27352667

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1556 ⋊ C₂
Sylow	C₅₃ ⋊ #17739

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1556	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂³×D₅₃	C₂²⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1556	D₅₃
hypocenter	C₅₃	#17739
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1556

950 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

728 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 138143);