#27358176

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2633 ⋊ C₂
Sylow	C₅₃ ⋊ #18314

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2633	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288124	C₂
hypercenter	#2633	D₅₃
hypocenter	C₅₃	#18314
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2633

958 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)(55,60)(59,68)(61,65)(62,69)(70,73,75,74)(71,77,78,82)(72,80,81,83)(76,84,85,79);(57,67)(61,65)(62,69)(63,66);(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)(55,59)(57,66)(58,64)(60,68)(61,69)(62,65)(63,67)(70,72)(71,76)(73,80)(74,83)(75,81)(77,84)(78,85)(79,82);(54,55,58,60)(56,62,64,69)(57,65,67,61)(59,66,68,63)(70,71)(72,79)(73,77)(74,82)(75,78)(76,80)(81,84)(83,85)

Polycyclic presentation:

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

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, 143652);