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C3×C5⋊D1235 #49150326

order37050
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C5

Aliases

direct productC3×C5⋊D1235
unambiguous semidirect productC5⋊(C3×D1235)

Splittings

Fitting, socle, HallC3×C52×C13×C19 ⋊ C2
derived, HallC52×C13×C19 ⋊ C2×C3
HallC3×C52×C19 ⋊ D13
HallC3×C52×C13 ⋊ D19
HallC3×C13×C19 ⋊ C5⋊D5
HallC52×C19 ⋊ C3×D13
HallC52×C13 ⋊ C3×D19
HallC13×C19 ⋊ C3×C5⋊D5
HallC3×C52 ⋊ D247
HallC3×C19 ⋊ C5⋊D65
HallC3×C13 ⋊ C5⋊D95
SylowC52 ⋊ C3×D247
SylowC19 ⋊ C3×C5⋊D65
SylowC13 ⋊ C3×C5⋊D95

Subgroups

subgroupG/H
center, hypercenterC3C5⋊D1235
commutator, cosocle, hypocenterC52×C13×C19C2×C3
FrattiniC1G
Fitting, socleC3×C52×C13×C19C2
LCSG ▷ C52×C13×C19
UCSC1 ◁ C3

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 18525

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

9267 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37050, 152);