Finite Group Databasebeta

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GO+(4,𝔽2)×ΓL(1,𝔽8)⋊C4 #72217776

order6048
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productGO+(4,𝔽2)×ΓL(1,𝔽8)⋊C4

Splittings

HallC32×ΓL(1,𝔽8) ⋊ C4×D4
SylowC7 ⋊ C3×C4×GO+(4,𝔽2)

Subgroups

subgroupG/H
center, Frattini, hypercenterC2AGL(1,𝔽7)×GO+(4,𝔽2)
commutatorC7×C3⋊D3C22×C3×C4
Fitting, socleC2×C32×C7C2×C3×D4
prime radicalC2×ΓL(1,𝔽8)×GO+(4,𝔽2)C2
cosocleC2×C7×C3⋊D3C23×C3
hypocenterC32×C7C3×C4×D4
LCSG ▷ C7×C3⋊D3 ▷ C32×C7
UCSC1 ◁ C2

656 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 91 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

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×AGL(1,𝔽7)×GO+(4,𝔽2)
maximal normal subgroup ofC4×AGL(1,𝔽7)×GO+(4,𝔽2)

GAP

Past the SmallGroups library coverage — no SmallGroups id.