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C3×ΓL(1,𝔽8)×H(5,𝔽2) #72244389

order2016
automorphism?
homologyAbelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3

Aliases

direct productC3×ΓL(1,𝔽8)×H(5,𝔽2)

Splittings

FittingC3×C7×H(5,𝔽2) ⋊ C3
HallC7×H(5,𝔽2) ⋊ C32
SylowC7 ⋊ C32×H(5,𝔽2)

Subgroups

subgroupG/H
centerC2×C3C24×ΓL(1,𝔽8)
commutator, cosocleC2×C7C24×C32
FrattiniC2C24×C3×ΓL(1,𝔽8)
FittingC3×C7×H(5,𝔽2)C3
socleC2×C3×C7C24×C3
hypercenterC3×H(5,𝔽2)ΓL(1,𝔽8)
hypocenterC7C32×H(5,𝔽2)
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C3×H(5,𝔽2)

996 conjugacy classes of subgroups; only the 544 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 544 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) = 168

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×H(5,𝔽2)×AGL(1,𝔽7), D3×ΓL(1,𝔽8)×H(5,𝔽2), H(5,𝔽2)×C3⋊AGL(1,𝔽7), …
maximal normal subgroup ofC3×H(5,𝔽2)×AGL(1,𝔽7), D3×ΓL(1,𝔽8)×H(5,𝔽2), H(5,𝔽2)×C3⋊AGL(1,𝔽7), …
maximal quotient ofΓL(1,𝔽8)×#6548⋊C3, ΓL(1,𝔽8)2×H(5,𝔽2), C3×ΓL(1,𝔽8)×#1004

GAP

Past the SmallGroups library coverage — no SmallGroups id.