Finite Group Databasebeta

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C7×#810 #10620608

order672
isoclinismisoclinic class of #213
automorphism?
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#810

Splittings

HallC3×C7 ⋊ #103
SylowC3 ⋊ C7×#103

Subgroups

subgroupG/H
centerC22×C7#67
commutatorC3×C4C2×C4×C7
FrattiniC2×C4C2×D3×C7
FittingC3×C7×C4⋊C4C2
socle, prime radicalC22×C3×C7D4
cosocleC2×C3×C4C22×C7
hypercenterC7×C4⋊C4D3
hypocenterC3C7×#103
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C7 ◁ C2×C4×C7 ◁ C7×C4⋊C4

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:

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13052565, #13052567, #13052569, …
maximal normal subgroup of#13052565, #13052567, #13052569, …
maximal quotient ofC7×#4452, C7×#4484, C7×#4486, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 190);