Finite Group Databasebeta

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#28319734

order13568
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

FittingC2×AGL(1,ℤ/8)×C53 ⋊ C4
SylowC53 ⋊ #20945

Subgroups

subgroupG/H
centerC2C2×#16811571
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×AGL(1,ℤ/8)×C53C4
socleC2×C53C2×#386
prime radicalC2×AGL(1,ℤ/8)×D53C2
cosocleC2×C4×D53C24
hypercenterC2×AGL(1,ℤ/8)#6157
hypocenterC53#20945
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 848

Polycyclic presentation:

2^8,53|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,i^a=i^23,e^b=eh,g^b=gh,i^b=i^30,f^c=fh,i^g=i^52

Conjugacy classes & character table

344 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1105210);