Finite Group Databasebeta

← Prev (#10621344) Next (#10621346) →

C7×#948 #10621345

order672
automorphism?
homologyAbelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22

Aliases

direct productC7×#948

Splittings

HallC3×C7 ⋊ #131
SylowC3 ⋊ C7×#131

Subgroups

subgroupG/H
centerC4×C7C22×D3
commutatorC2×C3C22×C4×C7
FrattiniC4C22×D3×C7
Fitting, prime radicalC3×C7×#41C2
socleC2×C3×C7C22×C4
cosocleC3×C4C23×C7
hypercenterC7×#41D3
hypocenterC3C7×#131
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C7×#41

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:

(1,2,3,4,5,6,7)(8,9,10)(13,19)(16,22)(20,25)(23,26);(11,13)(12,16)(14,20)(15,19)(17,23)(18,22)(21,25)(24,26);(9,10)(11,12,14,17,15,18,21,24)(13,16,20,23,19,22,25,26)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13054811, #13054812, #13054813, …
maximal normal subgroup of#13054811, #13054812, #13054813, …
maximal quotient ofC7×#4997, C7×#4999, C7×#5000, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 927);