Finite Group Databasebeta

← Prev (#49141682) Next (#49141684) →

C2×C3×C5×D609 #49141683

order36540
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×C5×D609
unambiguous semidirect product(C2×C3)⋊(C3×C5×D203)
unambiguous semidirect product(C2×C29)⋊(C3×C5×D21)

Splittings

Fitting, socleC2×C32×C5×C7×C29 ⋊ C2
derivedC3×C7×C29 ⋊ C22×C3×C5
HallC32×C5×C7×C29 ⋊ C22
HallC32×C7×C29 ⋊ C22×C5
HallC32×C5×C29 ⋊ C2×D7
HallC5×C7×C29 ⋊ C2×C3×D3
HallC32×C5×C7 ⋊ C2×D29
HallC32×C29 ⋊ C2×C5×D7
HallC7×C29 ⋊ C2×C3×C5×D3
HallC5×C29 ⋊ C2×C3×D21
HallC32×C7 ⋊ C2×C5×D29
HallC32×C5 ⋊ C2×D203
HallC5×C7 ⋊ C2×C3×D87
SylowC29 ⋊ C2×C3×C5×D21
SylowC32 ⋊ C2×C5×D203
SylowC7 ⋊ C2×C3×C5×D87

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5D609
commutator, cosocle, hypocenterC3×C7×C29C22×C3×C5
FrattiniC1G
Fitting, socleC2×C32×C5×C7×C29C2
LCSG ▷ C3×C7×C29
UCSC1 ◁ C2×C3×C5

176 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 49
transitive degreeμt(G) = 12180

A minimal faithful permutation representation — 2 generators on 49 points:

(3,4,5)(7,8)(9,10,11,12,13)(15,16)(17,18)(19,20)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49);(1,2)(6,7,8)(14,15,17,19,20,18,16)(21,22,24,26,28,30,32,34,36,38,40,42,44,46,48,49,47,45,43,41,39,37,35,33,31,29,27,25,23)

Conjugacy classes & character table

9180 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 525);