Finite Group Databasebeta

← Prev (#67387) Next (#67389) →

Cayley diagram

C7×#272 #67388

order378
isoclinismisoclinic class of #272
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×#272

Splittings

Fitting, HallC7×H(3,𝔽3) ⋊ C2
derived, SylowH(3,𝔽3) ⋊ C2×C7

Subgroups

subgroupG/H
center, socle, hypercenterC3×C7C3⋊D3
commutator, cosocle, hypocenterH(3,𝔽3)C2×C7
FrattiniC3C7×C3⋊D3
FittingC7×H(3,𝔽3)C2
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C3×C7

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 63

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2,3,4,5,6,7)(9,11)(12,16)(13,15);(9,12,13)(11,15,16);(8,9,11)(10,13,15)(12,16,14)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

derived subgroup of(((C3⋊D3)⋊C3)⋊C2)⋊D7, #1088⋊D7, C7⋊SU(3,𝔽2)
maximal subgroup ofC7×#1088, C7⋊#1088, D7×#272, …
maximal normal subgroup ofC7×#1088, C7⋊#1088, D7×#272, …
maximal quotient ofC7×#1084, C2×C7×#272, C7×#4181, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 41);