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Cayley diagram

3−1+2⋊D7 #67386

order378
isoclinismisoclinic class of 3−1+2⋊D7
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product3−1+2⋊D7

Splittings

Fitting, HallC7×3−1+2 ⋊ C2
derivedC7×C9 ⋊ C2×C3
Sylow3−1+2 ⋊ D7
SylowC7 ⋊ AGL(1,ℤ/9)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC7×C9C2×C3
FrattiniC3C3×D21
FittingC7×3−1+2C2
socleC3×C7C3×D3
LCSG ▷ C7×C9
UCSC1

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 63

A minimal faithful permutation representation — 2 generators on 16 points:

(2,3)(4,5)(6,7)(9,13,16,11,12,15)(10,14);(1,2,4,6,7,5,3)(8,16,11,14,12,13,10,9,15)

Polycyclic presentation:

2,3^2,7,3|c^3=e^2,c^a=c^2e,d^a=d^6,e^a=e^2,c^b=ce

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×AGL(1,ℤ/9), C2×3−1+2⋊D7, C7⋊#4183, …
maximal normal subgroup ofD7×AGL(1,ℤ/9), C2×3−1+2⋊D7, ΓL(1,𝔽8)⋊AGL(1,ℤ/9), …
maximal quotient of3−1+2⋊Dic7, C2×3−1+2⋊D7, #11765673, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 39);