Finite Group Databasebeta

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C3⋊(C7⋊(C7⋊#63121)) #50304425

order38220
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C7

Aliases

unambiguous semidirect productC3⋊(C7⋊(C7⋊#63121))
unambiguous semidirect productC7⋊(C3⋊(C7⋊#63121))

Splittings

Fitting, socle, derived, HallC3×C5×C72×C13 ⋊ C4
HallC5×C72×C13 ⋊ Dic3
HallC3×C72×C13 ⋊ AGL(1,𝔽5)
HallC3×C5×C72 ⋊ #261
HallC72×C13 ⋊ ΓL(1,𝔽16)
HallC5×C72 ⋊ C3⋊#261
HallC3×C5×C13 ⋊ C7⋊Dic7
HallC3×C72 ⋊ #63121
HallC5×C13 ⋊ C3⋊(C7⋊Dic7)
SylowC72 ⋊ C3⋊#63121
HallC3×C13 ⋊ C7⋊(C7⋊AGL(1,𝔽5))
HallC3×C5 ⋊ C7⋊(C7⋊#261)
SylowC13 ⋊ C3⋊(C7⋊(C7⋊AGL(1,𝔽5)))
SylowC5 ⋊ C3⋊(C7⋊(C7⋊#261))
SylowC3 ⋊ C7⋊(C7⋊#63121)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C72×C13C4
prime radical, cosocleC3×C72×D65C2
LCSG ▷ C3×C5×C72×C13
UCSC1

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 9555

Polycyclic presentation:

2^2,3,5,7^2,13|a^2=b,c^a=c^2,d^a=d^2,e^a=e^6,f^a=f^6,g^a=g^8,d^b=d^4,g^b=g^12

Conjugacy classes & character table

2502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 1111);