#27362192

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1632 ⋊ C₂
Sylow	C₅₃ ⋊ #18711

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1632	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#13133188	D₄
hypercenter	#1632	D₅₃
hypocenter	C₅₃	#18711
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1632

818 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 147668);