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#6398⋊C3 #10620452

order672
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#6398⋊C3

Splittings

Hall#6398 ⋊ C3
HallΓL(1,𝔽8) ⋊ #101
SylowC7 ⋊ C3×#101

Subgroups

subgroupG/H
centerC2(C2×AGL(1,𝔽7))⋊C4
commutatorC22×C7C2×C3×C4
FrattiniC2×C4C2×AGL(1,𝔽7)
FittingC7×M16C2×C3
socleC2×C7C3×C22⋊C4
prime radicalC22×ΓL(1,𝔽8)C2×C4
cosocleC2×C4×C7C22×C3
hypercenterM16AGL(1,𝔽7)
hypocenterC7C3×#101
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Polycyclic presentation:

2^2,3,2^3,7|a^2=f,b^2=d,d^2=f,b^a=be,d^a=df,g^a=g^6,e^b=ef,g^c=g^4

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89517⋊C3, #89520⋊C3, #101×AGL(1,𝔽7), …
maximal normal subgroup of#89517⋊C3, #89520⋊C3, #101×AGL(1,𝔽7), …
maximal quotient of#89260⋊C3, #89264⋊C3, #89277⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 34);