Finite Group Databasebeta

← Prev (#89131) Next (#89133) →

Cayley diagram

D217 #89132

order434
isoclinismisoclinic class of D217
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD217
unambiguous semidirect productC7⋊D31
unambiguous semidirect productC31⋊D7

Splittings

Fitting, socle, derived, HallC7×C31 ⋊ C2
SylowC31 ⋊ D7
SylowC7 ⋊ D31

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C31C2
LCSG ▷ C7×C31
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 217

A minimal faithful permutation representation — 2 generators on 38 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38);(2,7)(3,6)(4,5)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Polycyclic presentation:

2,7,31|b^a=b^6,c^a=c^30

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×D31, C2×D217, (C31⋊C3)⋊D7, …
maximal normal subgroup ofD7×D31, C2×D217, (C31⋊C3)⋊D7, …
maximal quotient ofDic217, C2×D217, C3×D217, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(434, 3);