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Cayley diagram

D175 #66837

order350
isoclinismisoclinic class of D175
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD175
unambiguous semidirect productC7⋊D25
unambiguous semidirect productC25⋊D7

Splittings

Fitting, derived, HallC7×C25 ⋊ C2
SylowC25 ⋊ D7
SylowC7 ⋊ D25

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC7×C25C2
FrattiniC5D35
socleC5×C7D5
LCSG ▷ C7×C25
UCSC1

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:

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

Polycyclic presentation:

2,5,7,5|b^5=d^4,b^a=b^4d,c^a=c^6,d^a=d^4

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×D25, C2×D175, ΓL(1,𝔽8)⋊D25, …
maximal normal subgroup ofD7×D25, C2×D175, ΓL(1,𝔽8)⋊D25, …
maximal quotient ofDic175, C2×D175, C3×D175, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(350, 3);