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C3×C5⋊D805 #37563409

order24150
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C5

Aliases

direct productC3×C5⋊D805
unambiguous semidirect productC5⋊(C3×D805)
unambiguous semidirect productC23⋊(C3×C5⋊D35)

Splittings

Fitting, socle, HallC3×C52×C7×C23 ⋊ C2
derived, HallC52×C7×C23 ⋊ C2×C3
HallC3×C52×C23 ⋊ D7
HallC52×C23 ⋊ C3×D7
HallC3×C52×C7 ⋊ D23
HallC3×C7×C23 ⋊ C5⋊D5
HallC52×C7 ⋊ C3×D23
HallC7×C23 ⋊ C3×C5⋊D5
HallC3×C52 ⋊ D161
HallC3×C23 ⋊ C5⋊D35
SylowC52 ⋊ C3×D161
SylowC23 ⋊ C3×C5⋊D35
HallC3×C7 ⋊ C5⋊D115
SylowC7 ⋊ C3×C5⋊D115

Subgroups

subgroupG/H
center, hypercenterC3C5⋊D805
commutator, cosocle, hypocenterC52×C7×C23C2×C3
FrattiniC1G
Fitting, socleC3×C52×C7×C23C2
LCSG ▷ C52×C7×C23
UCSC1 ◁ C3

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 12075

A minimal faithful permutation representation — 3 generators on 43 points:

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

Conjugacy classes & character table

6042 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24150, 68);