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ΓL(1,𝔽8)×#1136 #72244994

order2352
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×#1136

Splittings

HallC7×#1136 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ M16
SylowC72 ⋊ C3×M16

Subgroups

subgroupG/H
center, FrattiniC4C2×D7×ΓL(1,𝔽8)
commutator, socleC2×C72C2×C3×C4
FittingC2×C4×C72C2×C3
prime radicalC22×C7×ΓL(1,𝔽8)C4
cosocleC4×C72C22×C3
hypercenterC2×C4D7×ΓL(1,𝔽8)
hypocenterC72C3×M16
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C4 ◁ C2×C4

88 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#72244996, ΓL(1,𝔽8)×#6495, #72245047, …
maximal normal subgroup of#72244996, ΓL(1,𝔽8)×#6495, #72245047, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.