Finite Group Databasebeta

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Cayley diagram

C7×D25 #66835

order350
isoclinismisoclinic class of D25
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×D25
unambiguous semidirect productC25⋊(C2×C7)

Splittings

Fitting, HallC7×C25 ⋊ C2
derived, SylowC25 ⋊ C2×C7

Subgroups

subgroupG/H
center, hypercenterC7D25
commutator, cosocle, hypocenterC25C2×C7
FrattiniC5C7×D5
FittingC7×C25C2
socleC5×C7D5
LCSG ▷ C25
UCSC1 ◁ C7

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 175

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,3,4,5,6,7)(9,11)(10,14)(12,17)(13,20)(15,23)(16,22)(18,25)(19,27)(21,29)(24,30)(26,31)(28,32);(8,13,28,27,11,10,21,31,17,16,15,30,25,18,24,23,22,12,26,29,14,9,19,32,20)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC7×#1035, C7⋊#1035, D7×D25, …
maximal normal subgroup ofC7×#1035, C7⋊#1035, D7×D25, …
maximal quotient ofC7×Dic25, C2×C7×D25, C3×C7×D25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(350, 1);