Finite Group Databasebeta

← Prev (#49114974) Next (#49114976) →

C7×D143⋊(C3⋊D3) #49114975

order36036
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2×C3

Aliases

direct productC7×D143⋊(C3⋊D3)
unambiguous semidirect productD143⋊(C7×C3⋊D3)

Splittings

Fitting, socle, HallC32×C7×C11×C13 ⋊ C22
derived, HallC32×C11×C13 ⋊ C22×C7
HallC7×C11×C13 ⋊ C2×C3⋊D3
HallC32×C7×C13 ⋊ C2×D11
HallC32×C7×C11 ⋊ C2×D13
HallC11×C13 ⋊ C2×C7×C3⋊D3
HallC32×C13 ⋊ C2×C7×D11
HallC32×C11 ⋊ C2×C7×D13
HallC7×C13 ⋊ C3⋊D3×D11
HallC7×C11 ⋊ C3⋊D3×D13
HallC32×C7 ⋊ D11×D13
SylowC13 ⋊ C7×C3⋊D3×D11
SylowC11 ⋊ C7×C3⋊D3×D13
SylowC32 ⋊ C7×D11×D13

Subgroups

subgroupG/H
center, hypercenterC7D143⋊(C3⋊D3)
commutator, cosocle, hypocenterC32×C11×C13C22×C7
FrattiniC1G
Fitting, socleC32×C7×C11×C13C22
LCSG ▷ C32×C11×C13
UCSC1 ◁ C7

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 37
transitive degreeμt(G) = 9009

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

2436 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36036, 643);