Finite Group Databasebeta

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Cayley diagram

C3×Q8×D7 #66711

order336
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×Q8×D7

Splittings

FittingC3×C7×Q8 ⋊ C2
HallC3×C7 ⋊ C2×Q8
SylowC7 ⋊ C2×C3×Q8

Subgroups

subgroupG/H
centerC2×C3C22×D7
commutator, cosocleC2×C7C23×C3
FrattiniC2C22×C3×D7
FittingC3×C7×Q8C2
socleC2×C3×C7C23
prime radicalC2×C3×D7C22
hypercenterC3×Q8D7
hypocenterC7C2×C3×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C3×Q8

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×#215, D7×#216, #10621075, …
maximal normal subgroup ofD7×#215, D7×#216, #10621075, …
maximal quotient ofC3×#6450, C3×#6451, C3×#6452, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 180);