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Cayley diagram

C3×C4×Dic7 #66596

order336
isoclinismisoclinic class of D7
automorphism?
homologyAbelianization (H1) = C3×C42 · Schur multiplier (H2) = C4

Aliases

direct productC3×C4×Dic7

Splittings

derived, SylowC7 ⋊ C3×C42
HallC3×C7 ⋊ C42

Subgroups

subgroupG/H
center, hypercenterC2×C3×C4D7
commutator, hypocenterC7C3×C42
FrattiniC22C2×C3×D7
FittingC2×C3×C4×C7C2
socle, prime radicalC22×C3×C7C22
cosocleC22×C7C22×C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C4

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 2 generators on 18 points:

(2,3)(4,5)(6,7)(12,15,14,13)(16,17,18);(1,2,4,6,7,5,3)(8,11,10,9)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3⋊C8×Dic7, #10620487, #10620514, …
maximal normal subgroup ofC3⋊C8×Dic7, #10620487, #10620514, …
maximal quotient ofC3×C4×C7⋊C8, C3×#6377, C3×C8×Dic7, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 65);