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#63396⋊C3 #11720504

order864
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#63396⋊C3

Splittings

Sylow3−1+2 ⋊ #104

Subgroups

subgroupG/H
centerC4#611⋊C3
commutatorC4×C9C2×C3×C4
FrattiniC2×C3×C4C2×C3×D3
Fitting#41×3−1+2C2
socleC2×C3C3×#217
prime radicalC32×#41D3
cosocleC2×C4×C9C22×C3
hypercenter#41AGL(1,ℤ/9)
hypocenterC9C3×#104
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 72

Polycyclic presentation:

2^2,3,2^3,3^2|a^2=d,d^2=f,e^2=f,g^3=h,b^a=be,e^a=ef,g^a=g^2h^2,h^a=h^2,e^b=ef,g^c=gh

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#104×AGL(1,ℤ/9), #10588501⋊C3, #10588824⋊C3, …
maximal normal subgroup of#104×AGL(1,ℤ/9), #10588501⋊C3, #10588824⋊C3, …
maximal quotient of#10588208⋊C3, D4⋊(3−1+2⋊C8), #10588212⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 471);