Finite Group Databasebeta

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Cayley diagram

C4⋊(C3⋊D21) #92724

order504
isoclinismisoclinic class of #92722
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2×C3

Aliases

unambiguous semidirect productC4⋊(C3⋊D21)

Splittings

FittingC32×C4×C7 ⋊ C2
HallC32×C7 ⋊ D4
SylowC32 ⋊ D28
SylowC7 ⋊ C4⋊(C3⋊D3)

Subgroups

subgroupG/H
center, FrattiniC2C2×C3⋊D21
commutator, socle, cosocleC2×C32×C7C22
FittingC32×C4×C7C2
hypercenterC4C3⋊D21
hypocenterC32×C7D4
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C4

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 252

A minimal faithful permutation representation — 3 generators on 17 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11759154, #11759155, #11759159, …
maximal normal subgroup of#11759154, #11759155, #11759159, …
maximal quotient of#11759305, #11759306, #11759307, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 150);