Finite Group Databasebeta

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C2×#72217715 #72217872

order6720
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#72217715

Splittings

FittingC23×C5×C7 ⋊ C2×C3×C4
HallC2×(C22⋊D7)⋊AGL(1,𝔽5) ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C2×C4×D4
HallC5×C7 ⋊ C2×C3×C4×D4
HallΓL(1,𝔽8) ⋊ C2×D4×AGL(1,𝔽5)
SylowC7 ⋊ C2×C3×D4×AGL(1,𝔽5)
SylowC5 ⋊ C2×((C22⋊D7)⋊C3)⋊C4

Subgroups

subgroupG/H
centerC22C2×AGL(1,𝔽5)×AGL(1,𝔽7)
commutatorC2×C5×C7C23×C3×C4
FrattiniC2C22×AGL(1,𝔽5)×AGL(1,𝔽7)
FittingC23×C5×C7C2×C3×C4
socleC22×C5×C7C22×C3×C4
prime radicalC2×D5×(C22⋊D7)⋊C3C2
cosocleC2×C7×D5C24×C3
hypercenterC23AGL(1,𝔽5)×AGL(1,𝔽7)
hypocenterC5×C7C2×C3×C4×D4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C22 ◁ C23

1704 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×AGL(1,𝔽5)×AGL(1,𝔽7), #72218606, #72218608, …
maximal normal subgroup ofC2×D4×AGL(1,𝔽5)×AGL(1,𝔽7), #72218606, #72218608, …
maximal quotient of#72218599, #72218606, #72218611, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.