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C23⋊((C13⋊C3)⋊D19) #46801797

order34086
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC23⋊((C13⋊C3)⋊D19)
unambiguous semidirect product(C13⋊C3)⋊D437

Splittings

Fitting, socle, derived, HallC13×C19×C23 ⋊ C2×C3
HallC19×C23×C13⋊C3 ⋊ C2
HallD5681 ⋊ C3
HallC23×C13⋊C3 ⋊ D19
HallC19×C13⋊C3 ⋊ D23
HallC19×C23 ⋊ D13⋊C3
HallC13×C23 ⋊ C3×D19
HallC13×C19 ⋊ C3×D23
HallC13⋊C3 ⋊ D437
SylowC23 ⋊ (C13⋊C3)⋊D19
SylowC19 ⋊ C23⋊(D13⋊C3)
SylowC13 ⋊ C3×D437

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC13×C19×C23C2×C3
LCSG ▷ C13×C19×C23
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 55
transitive degreeμt(G) = 5681

A minimal faithful permutation representation — 2 generators on 55 points:

(2,6,7,3,4,8)(5,12,11,9,10,13)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55);(1,2,5,4,8,11,10,12,13,7,6,9,3)(14,15,17,19,21,23,25,27,29,31,32,30,28,26,24,22,20,18,16)(33,34,36,38,40,42,44,46,48,50,52,54,55,53,51,49,47,45,43,41,39,37,35)

Polycyclic presentation:

2,3,13,19,23|c^a=c^12,d^a=d^18,e^a=e^22,c^b=c^3

Conjugacy classes & character table

1534 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34086, 46);