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C5×D7×D153 #36339244

order21420
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D7×D153
unambiguous semidirect product(C9×D7)⋊(C5×D17)
unambiguous semidirect product(D7×C17)⋊(C5×D9)

Splittings

Fitting, HallC5×C7×C9×C17 ⋊ C22
derived, HallC7×C9×C17 ⋊ C22×C5
HallC5×C9×C17 ⋊ C2×D7
HallC5×C7×C17 ⋊ C2×D9
HallC5×C7×C9 ⋊ C2×D17
HallC9×C17 ⋊ C2×C5×D7
HallC7×C17 ⋊ C2×C5×D9
HallC5×C17 ⋊ D7×D9
HallC7×C9 ⋊ C2×C5×D17
HallC5×C9 ⋊ D7×D17
HallC5×C7 ⋊ C2×D153
SylowC17 ⋊ C5×D7×D9
SylowC9 ⋊ C5×D7×D17
SylowC7 ⋊ C2×C5×D153

Subgroups

subgroupG/H
center, hypercenterC5D7×D153
commutator, cosocle, hypocenterC7×C9×C17C22×C5
FrattiniC3C5×D7×D51
FittingC5×C7×C9×C17C22
socleC3×C5×C7×C17C2×D3
LCSG ▷ C7×C9×C17
UCSC1 ◁ C5

120 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 38
transitive degreeμt(G) = 5355

A minimal faithful permutation representation — 2 generators on 38 points:

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

Conjugacy classes & character table

1950 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 76);