Finite Group Databasebeta

← Prev (#90686) Next (#90688) →

Cayley diagram

(C19⋊C3)⋊Q8 #90687

order456
isoclinismisoclinic class of (C19⋊C3)⋊Q8
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product(C19⋊C3)⋊Q8
unambiguous semidirect productDic38⋊C3

Splittings

HallDic38 ⋊ C3
HallC19⋊C3 ⋊ Q8
SylowC19 ⋊ C3×Q8

Subgroups

subgroupG/H
center, FrattiniC2C2×D19⋊C3
commutator, socle, cosocleC2×C19C22×C3
FittingC4×C19C2×C3
prime radicalC2×C19⋊C3C22
hypercenterC4D19⋊C3
hypocenterC19C3×Q8
LCSG ▷ C2×C19 ▷ C19
UCSC1 ◁ C2 ◁ C4

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 152

A minimal faithful permutation representation — 2 generators on 27 points:

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

Polycyclic presentation:

2^2,3,2,19|a^2=d,b^2=d,b^a=bd,e^a=e^18,e^c=e^7

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64516⋊C3, Dic76⋊C3, #64526⋊C3, …
maximal normal subgroup of#64516⋊C3, Dic76⋊C3, #64526⋊C3, …
maximal quotient of((C19⋊C3)⋊C4)⋊C4, C4⋊((C19⋊C3)⋊C4), C2×(C19⋊C3)⋊Q8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 7);