Finite Group Databasebeta

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C3×C5×D3×D187 #46778083

order33660
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC3×C5×D3×D187
unambiguous semidirect product(D3×C17)⋊(C3×C5×D11)

Splittings

Fitting, socle, HallC32×C5×C11×C17 ⋊ C22
derivedC3×C11×C17 ⋊ C22×C3×C5
HallC32×C11×C17 ⋊ C22×C5
HallC5×C11×C17 ⋊ C2×C3×D3
HallC32×C5×C17 ⋊ C2×D11
HallC32×C5×C11 ⋊ C2×D17
HallC11×C17 ⋊ C2×C3×C5×D3
HallC32×C17 ⋊ C2×C5×D11
HallC32×C11 ⋊ C2×C5×D17
HallC5×C17 ⋊ C3×D3×D11
HallC5×C11 ⋊ C3×D3×D17
HallC32×C5 ⋊ C2×D187
SylowC17 ⋊ C3×C5×D3×D11
SylowC11 ⋊ C3×C5×D3×D17
SylowC32 ⋊ C2×C5×D187

Subgroups

subgroupG/H
center, hypercenterC3×C5D3×D187
commutator, cosocle, hypocenterC3×C11×C17C22×C3×C5
FrattiniC1G
Fitting, socleC32×C5×C11×C17C22
LCSG ▷ C3×C11×C17
UCSC1 ◁ C3×C5

176 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 39
transitive degreeμt(G) = 5610

A minimal faithful permutation representation — 2 generators on 39 points:

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

Conjugacy classes & character table

4275 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33660, 251);