Finite Group Databasebeta

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AGL(1,𝔽7)×C3⋊S4 #72222258

order3024
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3

Aliases

direct productAGL(1,𝔽7)×C3⋊S4

Splittings

Fitting, socleC22×C3×C7 ⋊ C2×C3×D3
derivedC3×C7×A4 ⋊ C22×C3
SylowC7 ⋊ C2×C3×C3⋊S4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC3×C7×A4C22×C3
Fitting, socleC22×C3×C7C2×C3×D3
LCSG ▷ C3×C7×A4
UCSC1

416 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.

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

Full subgroup data →

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×A4⋊S4, D3×S4×AGL(1,𝔽7), C2×AGL(1,𝔽7)×C3⋊S4
maximal normal subgroup ofD3×S4×AGL(1,𝔽7), C2×AGL(1,𝔽7)×C3⋊S4
maximal quotient ofAGL(1,𝔽7)×A4⋊S4, C2×AGL(1,𝔽7)×C3⋊S4, AGL(1,𝔽7)×C3⋊GL(2,𝔽3)

GAP

Past the SmallGroups library coverage — no SmallGroups id.