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C5×(C3⋊D3)⋊D209 #49178709

order37620
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×(C3⋊D3)⋊D209
unambiguous semidirect product(C3⋊D3)⋊(C5×D209)

Splittings

Fitting, socle, HallC32×C5×C11×C19 ⋊ C22
derived, HallC32×C11×C19 ⋊ C22×C5
HallC5×C11×C19 ⋊ D32
HallC32×C5×C19 ⋊ C2×D11
HallC32×C5×C11 ⋊ C2×D19
HallC11×C19 ⋊ C5×D32
HallC32×C19 ⋊ C2×C5×D11
HallC32×C11 ⋊ C2×C5×D19
HallC5×C19 ⋊ (C3⋊D3)⋊D11
HallC5×C11 ⋊ (C3⋊D3)⋊D19
HallC32×C5 ⋊ C2×D209
SylowC19 ⋊ C5×(C3⋊D3)⋊D11
SylowC11 ⋊ C5×(C3⋊D3)⋊D19
SylowC32 ⋊ C2×C5×D209

Subgroups

subgroupG/H
center, hypercenterC5(C3⋊D3)⋊D209
commutator, cosocle, hypocenterC32×C11×C19C22×C5
FrattiniC1G
Fitting, socleC32×C5×C11×C19C22
LCSG ▷ C32×C11×C19
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) = 41
transitive degreeμt(G) = 6270

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

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

Conjugacy classes & character table

3165 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37620, 418);