Finite Group Databasebeta

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D3×C7×D345 #44394602

order28980
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productD3×C7×D345
unambiguous semidirect product(C3×D3)⋊(C7×D115)
unambiguous semidirect product(C5×D3)⋊(C7×D69)
unambiguous semidirect product(D3×C23)⋊(C7×D15)

Splittings

Fitting, socle, HallC32×C5×C7×C23 ⋊ C22
derived, HallC32×C5×C23 ⋊ C22×C7
HallC32×C7×C23 ⋊ C2×D5
HallC5×C7×C23 ⋊ D32
HallC32×C5×C7 ⋊ C2×D23
HallC32×C23 ⋊ C2×C7×D5
HallC7×C23 ⋊ D3×D15
HallC5×C23 ⋊ D32×C7
HallC32×C7 ⋊ C2×D115
HallC32×C5 ⋊ C2×C7×D23
HallC5×C7 ⋊ D3×D69
SylowC23 ⋊ D3×C7×D15
SylowC32 ⋊ C2×C7×D115
SylowC5 ⋊ D3×C7×D69

Subgroups

subgroupG/H
center, hypercenterC7D3×D345
commutator, cosocle, hypocenterC32×C5×C23C22×C7
FrattiniC1G
Fitting, socleC32×C5×C7×C23C22
LCSG ▷ C32×C5×C23
UCSC1 ◁ C7

176 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 41
transitive degreeμt(G) = 4830

A minimal faithful permutation representation — 2 generators on 41 points:

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

Conjugacy classes & character table

3654 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28980, 271);