Finite Group Databasebeta

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C23×D5×AGL(1,𝔽7) #72217632

order3360
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C210

Aliases

direct productC23×D5×AGL(1,𝔽7)

Splittings

Fitting, socleC23×C5×C7 ⋊ C22×C3
derived, HallC5×C7 ⋊ C25×C3
HallC23×D5×D7 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C25
HallΓL(1,𝔽8) ⋊ C24×D5
SylowC7 ⋊ C24×C3×D5
SylowC5 ⋊ C24×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC23D5×AGL(1,𝔽7)
commutator, cosocle, hypocenterC5×C7C25×C3
FrattiniC1G
Fitting, socleC23×C5×C7C22×C3
LCSG ▷ C5×C7
UCSC1 ◁ C23

2992 conjugacy classes of subgroups; only the 965 normal subgroups are shown.

normal subgroups only

This group has 2992 conjugacy classes of subgroups — too many to draw. Only its 965 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

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC23×AGL(1,𝔽5)×AGL(1,𝔽7), C2×D4×D5×AGL(1,𝔽7), C2×AGL(1,𝔽7)×C22⋊AGL(1,𝔽5), …
maximal normal subgroup ofC23×AGL(1,𝔽5)×AGL(1,𝔽7), C2×D4×D5×AGL(1,𝔽7), C2×AGL(1,𝔽7)×C22⋊AGL(1,𝔽5), …
maximal quotient ofC2×D4×D5×AGL(1,𝔽7), C24×D5×AGL(1,𝔽7), D5×#41×AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.