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C5×D3×D413 #37613397

order24780
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D3×D413
unambiguous semidirect product(D3×C7)⋊(C5×D59)
unambiguous semidirect product(D3×C59)⋊(C5×D7)

Splittings

Fitting, socle, HallC3×C5×C7×C59 ⋊ C22
derived, HallC3×C7×C59 ⋊ C22×C5
HallC5×C7×C59 ⋊ C2×D3
HallC3×C5×C59 ⋊ C2×D7
HallC7×C59 ⋊ C2×C5×D3
HallC5×C59 ⋊ D3×D7
HallC3×C59 ⋊ C2×C5×D7
HallC3×C5×C7 ⋊ C2×D59
SylowC59 ⋊ C5×D3×D7
HallC5×C7 ⋊ D3×D59
HallC3×C7 ⋊ C2×C5×D59
HallC3×C5 ⋊ C2×D413
SylowC7 ⋊ C5×D3×D59
SylowC3 ⋊ C2×C5×D413

Subgroups

subgroupG/H
center, hypercenterC5D3×D413
commutator, cosocle, hypocenterC3×C7×C59C22×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C59C22
LCSG ▷ C3×C7×C59
UCSC1 ◁ C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 74
transitive degreeμt(G) = 6195

A minimal faithful permutation representation — 2 generators on 74 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)(61,62)(63,64)(65,66)(67,68,69)(70,71,72,73,74);(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)(60,61,63,65,66,64,62)(68,69)

Conjugacy classes & character table

3120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24780, 59);