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Cayley diagram

C7×C3⋊Dic3 #6984

order252
isoclinismisoclinic class of C3⋊D3
automorphismAut(G) = C22×C3×AGL(2,𝔽3) (order 5184) · Inn(G) = C3⋊D3 · Out(G) = C22×C3×S4
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C3

Aliases

direct productC7×C3⋊Dic3
unambiguous semidirect productC3⋊(C7×Dic3)

Splittings

derived, SylowC32 ⋊ C4×C7
HallC32×C7 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2×C7C3⋊D3
commutator, hypocenterC32C4×C7
FrattiniC2C7×C3⋊D3
Fitting, socle, prime radicalC2×C32×C7C2
cosocleC2×C32C2×C7
LCSG ▷ C32
UCSC1 ◁ C2×C7

36 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:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11759403, #11759404, #11759405, …
maximal subgroup ofC7×#622, C7⋊#622, D7×C3⋊Dic3, …
maximal normal subgroup ofC7×#622, C7⋊#622, D7×C3⋊Dic3, …
maximal quotient ofC7×C3⋊(C3⋊C8), C2×C7×C3⋊Dic3, C7×C3⋊Dic9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 23);