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ΓL(1,𝔽8)×Dic67 #19150586

order5628
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×Dic67

Splittings

derived, HallC7×C67 ⋊ C3×C4
HallC7×Dic67 ⋊ C3
HallΓL(1,𝔽8)×C67 ⋊ C4
SylowC67 ⋊ C4×ΓL(1,𝔽8)
SylowC7 ⋊ C3×Dic67

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×D67
commutator, hypocenterC7×C67C3×C4
Fitting, socleC2×C7×C67C2×C3
LCSG ▷ C7×C67
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

350 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×(C67⋊C3)⋊C4, C3×ΓL(1,𝔽8)×Dic67, ΓL(1,𝔽8)×Dic201, …
maximal normal subgroup ofΓL(1,𝔽8)×(C67⋊C3)⋊C4, C3×ΓL(1,𝔽8)×Dic67, C5×ΓL(1,𝔽8)×Dic67
maximal quotient ofC7⋊C9×Dic67, C3×ΓL(1,𝔽8)×Dic67, ΓL(1,𝔽8)×Dic201, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(5628, 14);