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C5×C3⋊D759 #36384792

order22770
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊D759
unambiguous semidirect productC3⋊(C5×D759)
unambiguous semidirect productC23⋊(C5×C3⋊D33)

Splittings

Fitting, socle, HallC32×C5×C11×C23 ⋊ C2
derived, HallC32×C11×C23 ⋊ C2×C5
HallC5×C11×C23 ⋊ C3⋊D3
HallC32×C5×C23 ⋊ D11
HallC32×C5×C11 ⋊ D23
HallC11×C23 ⋊ C5×C3⋊D3
HallC32×C23 ⋊ C5×D11
HallC5×C23 ⋊ C3⋊D33
HallC32×C11 ⋊ C5×D23
HallC5×C11 ⋊ C3⋊D69
HallC32×C5 ⋊ D253
SylowC23 ⋊ C5×C3⋊D33
SylowC11 ⋊ C5×C3⋊D69
SylowC32 ⋊ C5×D253

Subgroups

subgroupG/H
center, hypercenterC5C3⋊D759
commutator, cosocle, hypocenterC32×C11×C23C2×C5
FrattiniC1G
Fitting, socleC32×C5×C11×C23C2
LCSG ▷ C32×C11×C23
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 11385

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

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

Conjugacy classes & character table

5700 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22770, 76);