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C7×#798 #10620596

order672
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C1

Aliases

direct productC7×#798

Splittings

FittingC3×C7×C16 ⋊ C2
HallC3×C7 ⋊ M32
SylowC3 ⋊ C7×M32

Subgroups

subgroupG/H
centerC7×C8C2×D3
commutatorC2×C3C2×C7×C8
FrattiniC8C2×D3×C7
FittingC3×C7×C16C2
socleC2×C3×C7C2×C8
prime radicalC2×D3×C7C8
cosocleC3×C8C22×C7
hypercenterC7×C16D3
hypocenterC3C7×M32
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C7×C8 ◁ C7×C16

52 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)(11,12,13,17,14,18,19,24,15,16,20,22,21,23,25,26);(9,10)(12,16)(17,22)(18,23)(24,26)

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×#798, #13052188, #13052190, …
maximal normal subgroup ofD7×#798, #13052188, #13052190, …
maximal quotient ofC7×Dic3⋊C16, C7×#4502, C7×(C2×D3)⋊C16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 178);