Finite Group Databasebeta

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C3×#6470 #10621218

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

Aliases

direct productC3×#6470

Splittings

FittingC3×C7×M16 ⋊ C2
HallC3×C7 ⋊ #131
SylowC7 ⋊ C3×#131

Subgroups

subgroupG/H
centerC3×C4C22×D7
commutatorC2×C7C22×C3×C4
FrattiniC4C22×C3×D7
FittingC3×C7×M16C2
socleC2×C3×C7C22×C4
prime radicalC3×#1157C2
cosocleC4×C7C23×C3
hypercenterC3×M16D7
hypocenterC7C3×#131
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C3×M16

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13053561, #13053565, #13053566, …
maximal normal subgroup of#13053561, #13053565, #13053566, …
maximal quotient ofC3×#89473, C3×#89478, C3×#89480, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 800);