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C7×#809 #10620607

order672
isoclinismisoclinic class of #213
automorphism?
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C22

Aliases

direct productC7×#809

Splittings

FittingC3×C7×C4⋊C4 ⋊ C2
HallC3×C7 ⋊ #102
SylowC3 ⋊ C7×#102

Subgroups

subgroupG/H
centerC22×C7#67
commutatorC3×C4C2×C4×C7
FrattiniC2×C4C2×D3×C7
FittingC3×C7×C4⋊C4C2
socleC22×C3×C7D4
prime radicalC2×C7×D12C2
cosocleC2×C3×C4C22×C7
hypercenterC7×C4⋊C4D3
hypocenterC3C7×#102
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C7 ◁ C2×C4×C7 ◁ C7×C4⋊C4

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13052541, #13052543, #13052545, …
maximal normal subgroup of#13052541, #13052543, #13052545, …
maximal quotient ofC7×#4451, C7×#4483, C7×#4485, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 189);