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C22×C3×AGL(1,𝔽8) #10621679

order672
isoclinismisoclinic class of AGL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2

Aliases

direct productC22×C3×AGL(1,𝔽8)
unambiguous semidirect product(C25×C3)⋊C7

Splittings

Fitting, socle, HallC25×C3 ⋊ C7
derivedC23 ⋊ C22×C3×C7
SylowC25 ⋊ C3×C7

Subgroups

subgroupG/H
center, hypercenterC22×C3AGL(1,𝔽8)
commutator, cosocle, hypocenterC23C22×C3×C7
FrattiniC1G
Fitting, socleC25×C3C7
LCSG ▷ C23
UCSC1 ◁ C22×C3

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 84

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

derived subgroup ofD3×C22⋊AΓL(1,𝔽8), C2×D3×C22⋊AΓL(1,𝔽8)
maximal subgroup ofC2×Dic3×AGL(1,𝔽8), #67×AGL(1,𝔽8), C2×C3×C4×AGL(1,𝔽8), …
maximal normal subgroup ofC2×Dic3×AGL(1,𝔽8), #67×AGL(1,𝔽8), C2×C3×C4×AGL(1,𝔽8), …
maximal quotient ofC2×C3×C4×AGL(1,𝔽8), C3×D4×AGL(1,𝔽8), C3×Q8×AGL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 1261);