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Cayley diagram

C3×C7⋊S4 #92749

order504
isoclinismisoclinic class of C7⋊S4
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×C7⋊S4
unambiguous semidirect productA4⋊(C3×D7)

Splittings

Fitting, socleC22×C3×C7 ⋊ D3
derivedC7×A4 ⋊ C2×C3
SylowC7 ⋊ C3×S4

Subgroups

subgroupG/H
center, hypercenterC3C7⋊S4
commutator, cosocle, hypocenterC7×A4C2×C3
FrattiniC1G
Fitting, socleC22×C3×C7D3
LCSG ▷ C7×A4
UCSC1 ◁ C3

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D7×S4, D3×C7⋊S4, D21⋊S4, …
maximal normal subgroup ofC3×D7×S4, D3×C7⋊S4, D21⋊S4, …
maximal quotient ofC3×C7⋊#226, C3×C7⋊GL(2,𝔽3), C3×C7⋊SL(2,ℤ/4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 175);