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C5×(C3⋊D3)⋊D203 #49141603

order36540
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×(C3⋊D3)⋊D203
unambiguous semidirect product(C3⋊D3)⋊(C5×D203)

Splittings

Fitting, socle, HallC32×C5×C7×C29 ⋊ C22
derived, HallC32×C7×C29 ⋊ C22×C5
HallC32×C5×C29 ⋊ C2×D7
HallC5×C7×C29 ⋊ D32
HallC32×C5×C7 ⋊ C2×D29
HallC32×C29 ⋊ C2×C5×D7
HallC7×C29 ⋊ C5×D32
HallC5×C29 ⋊ (C3⋊D3)⋊D7
HallC32×C7 ⋊ C2×C5×D29
HallC32×C5 ⋊ C2×D203
HallC5×C7 ⋊ (C3⋊D3)⋊D29
SylowC29 ⋊ C5×(C3⋊D3)⋊D7
SylowC32 ⋊ C2×C5×D203
SylowC7 ⋊ C5×(C3⋊D3)⋊D29

Subgroups

subgroupG/H
center, hypercenterC5(C3⋊D3)⋊D203
commutator, cosocle, hypocenterC32×C7×C29C22×C5
FrattiniC1G
Fitting, socleC32×C5×C7×C29C22
LCSG ▷ C32×C7×C29
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) = 47
transitive degreeμt(G) = 6090

A minimal faithful permutation representation — 2 generators on 47 points:

(1,2,3,4,5)(7,8)(9,10)(12,13,15,17,18,16,14)(19,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21);(6,7,8)(10,11)(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)(42,43)(44,45)(46,47)

Conjugacy classes & character table

3075 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 445);