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C3×#6409 #10620592

order672
isoclinismisoclinic class of #1141
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#6409

Splittings

HallC3×C7 ⋊ #103
SylowC7 ⋊ C3×#103

Subgroups

subgroupG/H
centerC22×C3C22⋊D7
commutatorC4×C7C2×C3×C4
FrattiniC2×C4C2×C3×D7
FittingC2×C3×C7×Q8C2
socle, prime radicalC22×C3×C7D4
cosocleC2×C4×C7C22×C3
hypercenterC2×C3×Q8D7
hypocenterC7C3×#103
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#6409, #13054571, #13054572, …
maximal normal subgroup ofD3×#6409, #13054571, #13054572, …
maximal quotient ofC3×#89324, C3×Q8⋊(C7⋊C8), C3×#89333, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 174);