Finite Group Databasebeta

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ΓL(1,𝔽8)×C22⋊Dic7 #72245874

order2352
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productΓL(1,𝔽8)×C22⋊Dic7

Splittings

HallC7×C22⋊Dic7 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C22⋊C4
SylowC72 ⋊ C3×C22⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C2×D7×ΓL(1,𝔽8)
commutatorC2×C72C2×C3×C4
FittingC23×C72C2×C3
socle, cosocleC22×C72C22×C3
prime radicalC23×C7×ΓL(1,𝔽8)C2
hypercenterC23D7×ΓL(1,𝔽8)
hypocenterC72C3×C22⋊C4
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C22 ◁ C23

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245886, #72245925, ΓL(1,𝔽8)×#6500, …
maximal normal subgroup of#72245886, #72245925, ΓL(1,𝔽8)×#6500, …
maximal quotient ofΓL(1,𝔽8)×#6407, ΓL(1,𝔽8)×#6408

GAP

Past the SmallGroups library coverage — no SmallGroups id.