Finite Group Databasebeta

← Prev (#11724766) Next (#11724768) →

Dic217 #11724767

order868
isoclinismisoclinic class of D217
automorphismAut(G) order 78120
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic217
unambiguous semidirect productC7⋊Dic31
unambiguous semidirect productC31⋊Dic7

Splittings

derived, HallC7×C31 ⋊ C4
SylowC31 ⋊ Dic7
SylowC7 ⋊ Dic31

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D217
commutator, hypocenterC7×C31C4
Fitting, socle, prime radical, cosocleC2×C7×C31C2
LCSG ▷ C7×C31
UCSC1 ◁ C2

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 868

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2)(3,4)(5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42);(1,3,2,4)(6,11)(7,10)(8,9)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic7×D31, D7×Dic31, #13181909, …
maximal normal subgroup ofDic7×D31, D7×Dic31, #13181909, …
maximal quotient ofC7⋊(C31⋊C8), C2×Dic217, C3×Dic217, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(868, 3);