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D3×C7×D255 #36339503

order21420
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productD3×C7×D255
unambiguous semidirect product(C3×D3)⋊(C7×D85)
unambiguous semidirect product(C5×D3)⋊(C7×D51)
unambiguous semidirect product(D3×C17)⋊(C7×D15)

Splittings

Fitting, socle, HallC32×C5×C7×C17 ⋊ C22
derived, HallC32×C5×C17 ⋊ C22×C7
HallC32×C7×C17 ⋊ C2×D5
HallC5×C7×C17 ⋊ D32
HallC32×C5×C7 ⋊ C2×D17
HallC32×C17 ⋊ C2×C7×D5
HallC7×C17 ⋊ D3×D15
HallC5×C17 ⋊ D32×C7
HallC32×C7 ⋊ C2×D85
HallC32×C5 ⋊ C2×C7×D17
HallC5×C7 ⋊ D3×D51
SylowC17 ⋊ D3×C7×D15
SylowC32 ⋊ C2×C7×D85
SylowC5 ⋊ D3×C7×D51

Subgroups

subgroupG/H
center, hypercenterC7D3×D255
commutator, cosocle, hypocenterC32×C5×C17C22×C7
FrattiniC1G
Fitting, socleC32×C5×C7×C17C22
LCSG ▷ C32×C5×C17
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) = 35
transitive degreeμt(G) = 3570

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

2709 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 335);