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C7×#901 #10621298

order672
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22

Aliases

direct productC7×#901

Splittings

FittingC2×C3×C7×C8 ⋊ C2
HallC3×C7 ⋊ #131
SylowC3 ⋊ C7×#131

Subgroups

subgroupG/H
centerC7×C8C2×D3
commutatorC2×C3C22×C4×C7
FrattiniC4C22×D3×C7
FittingC2×C3×C7×C8C2
socleC2×C3×C7C22×C4
prime radicalC7×#235C2
cosocleC3×C4C23×C7
hypercenterC2×C7×C8D3
hypocenterC3C7×#131
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C7×C8 ◁ C2×C7×C8

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)(11,13,14,19,15,20,21,25)(12,17,18,24,16,22,23,26);(12,16)(17,22)(18,23)(24,26);(9,10)(11,12)(13,17)(14,18)(15,16)(19,24)(20,22)(21,23)(25,26)

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13050915, #13050916, C7×#4508, …
maximal normal subgroup of#13050915, #13050916, C7×#4508, …
maximal quotient ofC7×C8×Dic6, C7×#4679, C7×C8×D12, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 880);