Finite Group Databasebeta

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Cayley diagram

C4⋊(C3⋊(C3⋊Dic3)) #88858

order432
isoclinismisoclinic class of #6308
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C33

Aliases

unambiguous semidirect productC4⋊(C3⋊(C3⋊Dic3))

Splittings

SylowC33 ⋊ C4⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C2×C3⋊(C3⋊D3)
commutatorC2×C33C2×C4
FittingC2×C33×C4C2
socle, prime radical, cosocleC22×C33C22
hypercenterC2×C4C3⋊(C3⋊D3)
hypocenterC33C4⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 432

A minimal faithful permutation representation — 4 generators on 17 points:

(5,6,7,8)(15,16,17);(1,2,3,4)(6,8)(10,11)(13,14)(16,17);(12,13,14);(9,10,11)

Polycyclic presentation:

2^4,3^3|a^2=c,b^2=d,b^a=bd,e^a=e^2,f^a=f^2,g^a=g^2

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11722032, #11722034, #11722038, …
maximal normal subgroup of#11722032, #11722034, #11722038, …
maximal quotient ofC4⋊(C3⋊(C3⋊(C3⋊C8))), #11722187, #11722188, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 505);