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C5⋊((C13⋊C3)⋊D59) #37527668

order23010
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊((C13⋊C3)⋊D59)
unambiguous semidirect product(C13⋊C3)⋊D295
unambiguous semidirect productC59⋊(C5⋊(D13⋊C3))
unambiguous semidirect productD3835⋊C3

Splittings

Fitting, socle, derived, HallC5×C13×C59 ⋊ C2×C3
HallC5×C13⋊C3×C59 ⋊ C2
HallD3835 ⋊ C3
HallC13⋊C3×C59 ⋊ D5
HallC13×C59 ⋊ C3×D5
HallC5×C59 ⋊ D13⋊C3
HallC5×C13⋊C3 ⋊ D59
HallC5×C13 ⋊ C3×D59
SylowC59 ⋊ C5⋊(D13⋊C3)
HallC13⋊C3 ⋊ D295
SylowC13 ⋊ C3×D295
SylowC5 ⋊ (C13⋊C3)⋊D59

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C13×C59C2×C3
LCSG ▷ C5×C13×C59
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 77
transitive degreeμt(G) = 3835

A minimal faithful permutation representation — 2 generators on 77 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(61,65,66,62,63,67)(64,71,70,68,69,72)(74,75)(76,77);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(60,61,64,63,67,70,69,71,72,66,65,68,62)(73,74,76,77,75)

Polycyclic presentation:

2,3,5,13,59|c^a=c^4,d^a=d^12,e^a=e^58,d^b=d^3

Conjugacy classes & character table

1037 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23010, 24);