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#6379⋊C3 #10620433

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

Aliases

unambiguous semidirect product#6379⋊C3

Splittings

FittingC42×C7 ⋊ C2×C3
Hall#6379 ⋊ C3
HallΓL(1,𝔽8) ⋊ #104
SylowC7 ⋊ C3×#104

Subgroups

subgroupG/H
centerC4C4⋊AGL(1,𝔽7)
commutatorC4×C7C2×C3×C4
FrattiniC2×C4C2×AGL(1,𝔽7)
FittingC42×C7C2×C3
socleC2×C7C3×C22⋊C4
prime radical#1157⋊C3C2
cosocleC2×C4×C7C22×C3
hypercenterC42AGL(1,𝔽7)
hypocenterC7C3×#104
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 56

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89472⋊C3, #89489⋊C3, #104×AGL(1,𝔽7), …
maximal normal subgroup of#89472⋊C3, #89489⋊C3, #104×AGL(1,𝔽7), …
maximal quotient of#89246⋊C3, #89247⋊C3, #89254⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 15);