Finite Group Databasebeta

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AGL(1,𝔽5)×C7⋊SL(2,𝔽3) #72217636

order3360
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productAGL(1,𝔽5)×C7⋊SL(2,𝔽3)

Splittings

Fitting, derivedC5×C7×Q8 ⋊ C3×C4
HallC7×Q8×AGL(1,𝔽5) ⋊ C3
HallQ8×AGL(1,𝔽5) ⋊ ΓL(1,𝔽8)
HallC5×C7 ⋊ C4×SL(2,𝔽3)
SylowC7 ⋊ AGL(1,𝔽5)×SL(2,𝔽3)
SylowC5 ⋊ C4×C7⋊SL(2,𝔽3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC2AGL(1,𝔽5)×C22⋊ΓL(1,𝔽8)
commutator, Fitting, hypocenterC5×C7×Q8C3×C4
socleC2×C5×C7C4×A4
prime radicalD5×C7⋊SL(2,𝔽3)C2
cosocleC7×Q8×D5C2×C3
LCSG ▷ C5×C7×Q8
UCSC1 ◁ C2

120 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 32 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

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×ΓL(1,𝔽8)×SL(2,𝔽3), AGL(1,𝔽5)×D7⋊SL(2,𝔽3)
maximal normal subgroup ofAGL(1,𝔽5)×ΓL(1,𝔽8)×SL(2,𝔽3), AGL(1,𝔽5)×D7⋊SL(2,𝔽3)

GAP

Past the SmallGroups library coverage — no SmallGroups id.