Finite Group Databasebeta

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C3×C7×#168 #11719704

order840
isoclinismisoclinic class of D5
automorphism?
homologyAbelianization (H1) = C3×C7×C8 · Schur multiplier (H2) = C1

Aliases

direct productC3×C7×#168

Splittings

derived, SylowC5 ⋊ C3×C7×C8
HallC3×C5×C7 ⋊ C8
HallC5×C7 ⋊ C3×C8
HallC3×C5 ⋊ C7×C8

Subgroups

subgroupG/H
center, hypercenterC3×C4×C7D5
commutator, hypocenterC5C3×C7×C8
FrattiniC4C3×C7×D5
FittingC3×C4×C5×C7C2
socle, prime radicalC2×C3×C5×C7C4
cosocleC4×C5C2×C3×C7
LCSG ▷ C5
UCSC1 ◁ C3×C4×C7

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 840

A minimal faithful permutation representation — 2 generators on 23 points:

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

Conjugacy classes & character table

336 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×C7×#674, C3×C7⋊#674, C7×C3⋊#674, …
maximal normal subgroup ofC3×C7×#674, C3×C7⋊#674, C7×C3⋊#674, …
maximal quotient ofC3×C7×#672, C2×C3×C7×#168

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 6);