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C23⋊((C19⋊C3)⋊D13) #46801796

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

Aliases

unambiguous semidirect productC23⋊((C19⋊C3)⋊D13)
unambiguous semidirect product(C19⋊C3)⋊D299

Splittings

Fitting, socle, derived, HallC13×C19×C23 ⋊ C2×C3
HallC13×C23×C19⋊C3 ⋊ C2
HallD5681 ⋊ C3
HallC23×C19⋊C3 ⋊ D13
HallC13×C19⋊C3 ⋊ D23
HallC19×C23 ⋊ C3×D13
HallC13×C23 ⋊ D19⋊C3
HallC13×C19 ⋊ C3×D23
HallC19⋊C3 ⋊ D299
SylowC23 ⋊ (C19⋊C3)⋊D13
SylowC19 ⋊ C3×D299
SylowC13 ⋊ C23⋊(D19⋊C3)

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:

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

Conjugacy classes & character table

1350 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34086, 45);