#27369224

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2969 ⋊ C₂
Sylow	C₅₃ ⋊ #19382

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133191
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2969	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#20286077	C₂
hypercenter	#2969	D₅₃
hypocenter	C₅₃	#19382
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2969

594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

1337 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 154700);