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C3×H(3,𝔽3)×AGL(1,𝔽7) #72249108

order3402
automorphism?
homologyAbelianization (H1) = C2×C34 · Schur multiplier (H2) = C37

Aliases

direct productC3×H(3,𝔽3)×AGL(1,𝔽7)

Splittings

FittingC3×C7×H(3,𝔽3) ⋊ C2×C3
HallC3×ΓL(1,𝔽8)×H(3,𝔽3) ⋊ C2
HallD7 ⋊ C32×H(3,𝔽3)
SylowC7 ⋊ C2×C32×H(3,𝔽3)

Subgroups

subgroupG/H
centerC32C32×AGL(1,𝔽7)
commutator, cosocleC3×C7C2×C34
FrattiniC3C33×AGL(1,𝔽7)
FittingC3×C7×H(3,𝔽3)C2×C3
socleC32×C7C2×C33
hypercenterC3×H(3,𝔽3)AGL(1,𝔽7)
hypocenterC7C2×C32×H(3,𝔽3)
LCSG ▷ C3×C7 ▷ C7
UCSC1 ◁ C32 ◁ C3×H(3,𝔽3)

1800 conjugacy classes of subgroups; only the 500 normal subgroups are shown.

normal subgroups only

This group has 1800 conjugacy classes of subgroups — too many to draw. Only its 500 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) = 189

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽7)×AΣL(1,𝔽27), AGL(1,𝔽7)×3−1+4, AGL(1,𝔽7)×#6909, …
maximal normal subgroup ofC3×AGL(1,𝔽7)×AΣL(1,𝔽27), AGL(1,𝔽7)×3−1+4, AGL(1,𝔽7)×#6909, …
maximal quotient ofC3×AGL(1,𝔽7)×AΣL(1,𝔽27), AGL(1,𝔽7)×#6909, AGL(1,𝔽7)×#6910, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.