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C5⋊((C13⋊C3)⋊(C7⋊D7)) #34062172

order19110
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C7

Aliases

unambiguous semidirect productC5⋊((C13⋊C3)⋊(C7⋊D7))
unambiguous semidirect product(C13⋊C3)⋊(C5⋊(C7⋊D7))

Splittings

Fitting, socle, derived, HallC5×C72×C13 ⋊ C2×C3
HallC5×C72×C13⋊C3 ⋊ C2
HallC5⋊(C7⋊D91) ⋊ C3
HallC72×C13⋊C3 ⋊ D5
HallC72×C13 ⋊ C3×D5
HallC5×C72 ⋊ D13⋊C3
HallC5×C13⋊C3 ⋊ C7⋊D7
HallC5×C13 ⋊ C3×C7⋊D7
SylowC72 ⋊ C5⋊(D13⋊C3)
HallC13⋊C3 ⋊ C5⋊(C7⋊D7)
SylowC13 ⋊ C3×C5⋊(C7⋊D7)
SylowC5 ⋊ (C13⋊C3)⋊(C7⋊D7)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C72×C13C2×C3
LCSG ▷ C5×C72×C13
UCSC1

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 3185

A minimal faithful permutation representation — 3 generators on 32 points:

(2,6,7,3,4,8)(5,12,11,9,10,13)(15,16)(17,18)(20,21)(22,23)(24,25)(27,28)(29,30)(31,32);(1,2,5,4,8,11,10,12,13,7,6,9,3)(14,15,17,18,16)(26,27,29,31,32,30,28);(19,20,22,24,25,23,21)

Polycyclic presentation:

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

Conjugacy classes & character table

862 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#50303808, #50303809, D7×C5⋊((C13⋊C3)⋊D7), …
maximal normal subgroup of#50303808, #50303809, D7×C5⋊((C13⋊C3)⋊D7), …
maximal quotient of#50303667, C2×C5⋊((C13⋊C3)⋊(C7⋊D7))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19110, 165);