Finite Group Databasebeta

← Prev (#45659489) Next (#45659491) →

C3×C5⋊(C5⋊(D73⋊C3)) #45659490

order32850
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C5

Aliases

direct productC3×C5⋊(C5⋊(D73⋊C3))
unambiguous semidirect productC5⋊(C3×C5⋊(D73⋊C3))
unambiguous semidirect product(C73⋊C3)⋊(C3×C5⋊D5)

Splittings

Fitting, socleC3×C52×C73 ⋊ C2×C3
derived, HallC52×C73 ⋊ C2×C32
HallC3×C52×C73⋊C3 ⋊ C2
HallC5⋊D365 ⋊ C32
HallC3×C73⋊C3 ⋊ C5⋊D5
SylowC73 ⋊ C32×C5⋊D5
SylowC52 ⋊ C3×D73⋊C3

Subgroups

subgroupG/H
center, hypercenterC3C5⋊(C5⋊(D73⋊C3))
commutator, cosocle, hypocenterC52×C73C2×C32
FrattiniC1G
Fitting, socleC3×C52×C73C2×C3
LCSG ▷ C52×C73
UCSC1 ◁ C3

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 86
transitive degreeμt(G) = 5475

A minimal faithful permutation representation — 3 generators on 86 points:

(5,6)(7,8)(10,11)(12,13)(15,19,20,16,17,21)(18,27,28,22,23,37)(24,40,41,30,31,54)(25,39,35,33,29,26)(32,59,60,43,44,73)(34,57,58,46,42,51)(36,38,56,48,49,52)(45,61,70,47,62,76)(50,55,75,63,64,71)(53,80,78,67,68,66)(65,74,81,69,77,85)(72,86,84,82,83,79);(1,2,3)(9,10,12,13,11)(14,15,18,24,32,45,46,33,17,21,26,36,50,65,67,49,23,29,37,51,66,79,82,64,31,42,52,54,70,81,86,77,44,61,68,71,73,60,75,80,62,59,74,83,85,76,41,56,57,40,55,72,84,78,58,28,39,27,38,53,69,63,48,35,20,19,25,34,47,43,30,22,16);(4,5,7,8,6)

Conjugacy classes & character table

1026 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32850, 93);