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C2×C3×AGL(1,𝔽5)×ΓL(1,𝔽8) #72217323

order2520
automorphism?
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×AGL(1,𝔽5)×ΓL(1,𝔽8)

Splittings

Fitting, socleC2×C3×C5×C7 ⋊ C3×C4
derived, HallC5×C7 ⋊ C2×C32×C4
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×C4
HallC2×C7×AGL(1,𝔽5) ⋊ C32
SylowC7 ⋊ C2×C32×AGL(1,𝔽5)
SylowC5 ⋊ C2×C3×C4×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC2×C3AGL(1,𝔽5)×ΓL(1,𝔽8)
commutator, hypocenterC5×C7C2×C32×C4
FrattiniC1G
Fitting, socleC2×C3×C5×C7C3×C4
prime radicalC2×C3×D5×ΓL(1,𝔽8)C2
cosocleC7×D5C22×C32
LCSG ▷ C5×C7
UCSC1 ◁ C2×C3

192 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 80 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

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C3×AGL(1,𝔽5)×AGL(1,𝔽7), C2×D3×AGL(1,𝔽5)×ΓL(1,𝔽8), C2×AGL(1,𝔽5)×C3⋊AGL(1,𝔽7), …
maximal normal subgroup ofC2×C3×AGL(1,𝔽5)×AGL(1,𝔽7), C2×D3×AGL(1,𝔽5)×ΓL(1,𝔽8), C2×AGL(1,𝔽5)×C3⋊AGL(1,𝔽7)
maximal quotient ofC3×D3×AGL(1,𝔽5)×ΓL(1,𝔽8), C2×AGL(1,𝔽5)×ΓL(1,𝔽8)2, AGL(1,𝔽5)×ΓL(1,𝔽8)×AGL(1,𝔽7), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.