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ΓL(1,𝔽8)×D32⋊C4 #72217554

order3024
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productΓL(1,𝔽8)×D32⋊C4

Splittings

HallC32×ΓL(1,𝔽8) ⋊ C22⋊C4
SylowC7 ⋊ C3×D32⋊C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×GO+(4,𝔽2)
commutatorC7×C3⋊D3C2×C3×C4
Fitting, socleC2×C32×C7C3×D4
prime radicalC2×D32×ΓL(1,𝔽8)C2
cosocleC2×C7×C3⋊D3C22×C3
hypocenterC32×C7C3×C22⋊C4
LCSG ▷ C7×C3⋊D3 ▷ C32×C7
UCSC1 ◁ C2

264 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 39 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 ofC4×ΓL(1,𝔽8)×GO+(4,𝔽2), #72217790, AGL(1,𝔽7)×D32⋊C4
maximal normal subgroup ofC4×ΓL(1,𝔽8)×GO+(4,𝔽2), #72217790, AGL(1,𝔽7)×D32⋊C4

GAP

Past the SmallGroups library coverage — no SmallGroups id.