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D7×5−1+2 #13182075

order1750
automorphismAut(G) = AGL(1,𝔽7)×#92539 (order 21000) · Inn(G) = C52×D7 · Out(G) = C3×AGL(1,𝔽5)
homologyAbelianization (H1) = C2×C52 · Schur multiplier (H2) = C1

Aliases

direct productD7×5−1+2
unambiguous semidirect productC7⋊(C2×5−1+2)
unambiguous semidirect product(D7×C25)⋊C5

Splittings

Fitting, HallC7×5−1+2 ⋊ C2
SylowC7 ⋊ C2×5−1+2

Subgroups

subgroupG/H
center, FrattiniC5C52×D7
commutator, socle, cosocleC5×C7C2×C52
FittingC7×5−1+2C2
prime radicalC52×D7C5
hypercenter5−1+2D7
hypocenterC7C2×5−1+2
LCSG ▷ C5×C7 ▷ C7
UCSC1 ◁ C5 ◁ 5−1+2

40 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)(8,9,11,15,22,10,13,17,24,29,14,20,26,23,30,21,19,16,25,31,28,12,18,27,32);(2,7)(3,6)(4,5)(9,12,19,20,13)(11,16,17,18,26)(15,23,27,24,25)(22,29,30,31,32)

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×5−1+2, D7×D25⋊C5
maximal normal subgroup ofAGL(1,𝔽7)×5−1+2, D7×D25⋊C5

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1750, 7);