Finite Group Databasebeta

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C3×#6511 #10621259

order672
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×#6511

Splittings

HallC3×C7 ⋊ #131
SylowC7 ⋊ C3×#131

Subgroups

subgroupG/H
centerC3×C4C22×D7
commutatorC2×C7C22×C3×C4
FrattiniC4C22×C3×D7
Fitting, prime radicalC3×C7×#41C2
socleC2×C3×C7C22×C4
cosocleC4×C7C23×C3
hypercenterC3×#41D7
hypocenterC7C3×#131
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C3×#41

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 3 generators on 26 points:

(1,2,3)(4,5,7,9,10,8,6)(13,19)(16,22)(20,25)(23,26);(11,13)(12,16)(14,20)(15,19)(17,23)(18,22)(21,25)(24,26);(5,6)(7,8)(9,10)(11,12,14,17,15,18,21,24)(13,16,20,23,19,22,25,26)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13054807, #13054808, #13054809, …
maximal normal subgroup of#13054807, #13054808, #13054809, …
maximal quotient ofC3×#89769, C3×#89771, C3×#89772, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 841);