Finite Group Databasebeta

← Prev (#50307661) Next (#50307663) →

C5×C3⋊((C61⋊C3)⋊D7) #50307662

order38430
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊((C61⋊C3)⋊D7)
unambiguous semidirect productC3⋊(C5×(C61⋊C3)⋊D7)

Splittings

Fitting, socleC3×C5×C7×C61 ⋊ C2×C3
derivedC3×C7×C61 ⋊ C2×C3×C5
HallC3×C5×C7×C61⋊C3 ⋊ C2
HallC3×C7×C61⋊C3 ⋊ C2×C5
HallC3×C5×C61⋊C3 ⋊ D7
HallC5×C7×C61 ⋊ C3×D3
HallC3×C61⋊C3 ⋊ C5×D7
HallC7×C61 ⋊ C3×C5×D3
HallC5×C61 ⋊ C3×D21
SylowC61 ⋊ C3×C5×D21
HallC5×C7 ⋊ C3⋊(D61⋊C3)
SylowC7 ⋊ C5×C3⋊(D61⋊C3)

Subgroups

subgroupG/H
center, hypercenterC5C3⋊((C61⋊C3)⋊D7)
commutator, cosocle, hypocenterC3×C7×C61C2×C3×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C61C2×C3
LCSG ▷ C3×C7×C61
UCSC1 ◁ C5

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 76
transitive degreeμt(G) = 6405

A minimal faithful permutation representation — 2 generators on 76 points:

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

Conjugacy classes & character table

1230 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38430, 171);