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C7×#822 #10620620

order672
isoclinismisoclinic class of #213
automorphism?
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×#822

Splittings

HallC3×C7 ⋊ #108
SylowC3 ⋊ C7×#108

Subgroups

subgroupG/H
centerC4×C7#67
commutatorC3×C4C2×C4×C7
FrattiniC2×C4C2×D3×C7
FittingC3×C7×M16C2
socleC2×C3×C7C4⋊C4
prime radicalC22×C3×C7Q8
cosocleC2×C3×C4C22×C7
hypercenterC7×M16D3
hypocenterC3C7×#108
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C7×M16

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×#822, #13053549, #13053550, …
maximal normal subgroup ofD7×#822, #13053549, #13053550, …
maximal quotient ofC7×#4479, C7×#4480, C7×#4558, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 202);