Finite Group Databasebeta

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Cayley diagram

C2×C4×D3×C7 #66716

order336
isoclinismisoclinic class of D3
automorphismAut(G) = C2×C3×D3×#120 (order 1152) · Inn(G) = D3 · Out(G) = C2×C3×#120
homologyAbelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4×D3×C7

Splittings

FittingC2×C3×C4×C7 ⋊ C2
derived, SylowC3 ⋊ C22×C4×C7
HallC3×C7 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4×C7D3
commutator, hypocenterC3C22×C4×C7
FrattiniC2C22×D3×C7
FittingC2×C3×C4×C7C2
socleC22×C3×C7C22
prime radicalC22×D3×C7C2
cosocleC2×C3C23×C7
LCSG ▷ C3
UCSC1 ◁ C2×C4×C7

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×D3)⋊(C7⋊C8), C7×(C2×D3)⋊C8, C2×D3×C7⋊C8, …
maximal normal subgroup of(C2×D3)⋊(C7⋊C8), C7×(C2×D3)⋊C8, C2×D3×C7⋊C8, …
maximal quotient ofC4×C7×Dic6, C42×D3×C7, C7×#872, …
automorphism group ofDic3×C29, C2×D3×C29, C22×C3×C29
outer automorphism group ofC22×C3×C29, #41×C29

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 185);