Finite Group Databasebeta

← Prev (#92768) Next (#92770) →

Cayley diagram

C2×C3×C7×A4 #92769

order504
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C3×D3×S4 (order 864) · Inn(G) = A4 · Out(G) = C22×C3×D3
homologyAbelianization (H1) = C2×C32×C7 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×C7×A4

Splittings

Fitting, socleC23×C3×C7 ⋊ C3
derivedC22 ⋊ C2×C32×C7
HallC23×C7 ⋊ C32
SylowC23 ⋊ C32×C7

Subgroups

subgroupG/H
center, hypercenterC2×C3×C7A4
commutator, cosocle, hypocenterC22C2×C32×C7
FrattiniC1G
Fitting, socleC23×C3×C7C3
LCSG ▷ C22
UCSC1 ◁ C2×C3×C7

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 126

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

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C7×SL(2,ℤ/4), C3×C7⋊SL(2,ℤ/4), C7×C3⋊SL(2,ℤ/4), …
maximal normal subgroup ofC3×C7×SL(2,ℤ/4), C3×C7⋊SL(2,ℤ/4), C7×C3⋊SL(2,ℤ/4), …
maximal quotient ofC3×C4×C7×A4, C2×C3×C7×SL(2,𝔽3), C3×C7×#41⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 195);