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C5×D51⋊D21 #36339497

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

Aliases

direct productC5×D51⋊D21
unambiguous semidirect productD51⋊(C5×D21)

Splittings

Fitting, socle, HallC32×C5×C7×C17 ⋊ C22
derived, HallC32×C7×C17 ⋊ C22×C5
HallC32×C5×C17 ⋊ C2×D7
HallC5×C7×C17 ⋊ D32
HallC32×C5×C7 ⋊ C2×D17
HallC32×C17 ⋊ C2×C5×D7
HallC7×C17 ⋊ C5×D32
HallC5×C17 ⋊ D3×D21
HallC32×C7 ⋊ C2×C5×D17
HallC32×C5 ⋊ D7×D17
HallC5×C7 ⋊ (C3⋊D3)⋊D17
SylowC17 ⋊ C5×D3×D21
SylowC32 ⋊ C5×D7×D17
SylowC7 ⋊ C5×(C3⋊D3)⋊D17

Subgroups

subgroupG/H
center, hypercenterC5D51⋊D21
commutator, cosocle, hypocenterC32×C7×C17C22×C5
FrattiniC1G
Fitting, socleC32×C5×C7×C17C22
LCSG ▷ C32×C7×C17
UCSC1 ◁ C5

176 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Conjugacy classes & character table

1500 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 329);