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C3×C5⋊D1265 #49182153

order37950
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C5

Aliases

direct productC3×C5⋊D1265
unambiguous semidirect productC5⋊(C3×D1265)
unambiguous semidirect productC11⋊(C3×C5⋊D115)
unambiguous semidirect productC23⋊(C3×C5⋊D55)

Splittings

Fitting, socle, HallC3×C52×C11×C23 ⋊ C2
derived, HallC52×C11×C23 ⋊ C2×C3
HallC3×C52×C23 ⋊ D11
HallC3×C52×C11 ⋊ D23
HallC3×C11×C23 ⋊ C5⋊D5
HallC52×C23 ⋊ C3×D11
HallC52×C11 ⋊ C3×D23
HallC11×C23 ⋊ C3×C5⋊D5
HallC3×C52 ⋊ D253
HallC3×C23 ⋊ C5⋊D55
HallC3×C11 ⋊ C5⋊D115
SylowC52 ⋊ C3×D253
SylowC23 ⋊ C3×C5⋊D55
SylowC11 ⋊ C3×C5⋊D115

Subgroups

subgroupG/H
center, hypercenterC3C5⋊D1265
commutator, cosocle, hypocenterC52×C11×C23C2×C3
FrattiniC1G
Fitting, socleC3×C52×C11×C23C2
LCSG ▷ C52×C11×C23
UCSC1 ◁ C3

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 47
transitive degreeμt(G) = 18975

A minimal faithful permutation representation — 3 generators on 47 points:

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

Conjugacy classes & character table

9492 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37950, 90);