Finite Group Databasebeta

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C3×AGL(1,𝔽5)×#157 #72216970

order2160
automorphism?
homologyAbelianization (H1) = C3×C42 · Schur multiplier (H2) = C3×C4

Aliases

direct productC3×AGL(1,𝔽5)×#157

Splittings

Fitting, socle, HallC33×C5 ⋊ C42
derivedC32×C5 ⋊ C3×C42
SylowC33 ⋊ C4×AGL(1,𝔽5)
SylowC5 ⋊ C3×C4×#157

Subgroups

subgroupG/H
center, hypercenterC3AGL(1,𝔽5)×#157
commutator, hypocenterC32×C5C3×C42
FrattiniC1G
Fitting, socleC33×C5C42
prime radicalC3×D5×C3⋊D3C22
cosocleD5×C3⋊D3C22×C3
LCSG ▷ C32×C5
UCSC1 ◁ C3

208 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 44 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

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽5)×AGL(1,𝔽9), C2×C3×AGL(1,𝔽5)×#157, AGL(1,𝔽5)×ΓL(1,𝔽8)×#157, …
maximal normal subgroup ofC3×AGL(1,𝔽5)×AGL(1,𝔽9), C2×C3×AGL(1,𝔽5)×#157, C3×AGL(1,𝔽5)×GO+(4,𝔽2), …
maximal quotient ofC2×C3×AGL(1,𝔽5)×#157, AGL(1,𝔽5)×ΓL(1,𝔽8)×#157, #157×ΓL(1,𝔽8)⋊AGL(1,𝔽5), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.