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Cayley diagram

D7⋊H(3,𝔽3) #67359

order378
isoclinismisoclinic class of #67355
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

unambiguous semidirect productD7⋊H(3,𝔽3)
unambiguous semidirect product(C3×AGL(1,𝔽7))⋊C3

Splittings

FittingC32×C7 ⋊ C2×C3
HallC7⋊H(3,𝔽3) ⋊ C2
HallD7 ⋊ H(3,𝔽3)
SylowC7 ⋊ C2×H(3,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC3C3×AGL(1,𝔽7)
commutator, socle, cosocleC3×C7C2×C32
FittingC32×C7C2×C3
hypercenterC32AGL(1,𝔽7)
hypocenterC7C2×H(3,𝔽3)
LCSG ▷ C3×C7 ▷ C7
UCSC1 ◁ C3 ◁ C32

44 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:

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

Polycyclic presentation:

2,3^3,7|e^a=e^6,c^b=cd,e^b=e^4

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7⋊((C3⋊D3)⋊C3), C2×D7⋊H(3,𝔽3), #11765586, …
maximal normal subgroup ofD7⋊((C3⋊D3)⋊C3), C2×D7⋊H(3,𝔽3), #11765586, …
maximal quotient ofDic7⋊H(3,𝔽3), C2×D7⋊H(3,𝔽3), #11765586, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 12);