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C3⋊(C3⋊(C19⋊(Dic11⋊C5))) #49178458

order37620
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊(C3⋊(C19⋊(Dic11⋊C5)))
unambiguous semidirect productC19⋊(C3⋊(C3⋊(Dic11⋊C5)))
unambiguous semidirect product(C11⋊C5)⋊(C3⋊Dic57)
unambiguous semidirect product(C3⋊Dic627)⋊C5

Splittings

derived, HallC32×C11×C19 ⋊ C4×C5
HallC32×C19×C11⋊C5 ⋊ C4
HallC3⋊Dic627 ⋊ C5
HallC19×C11⋊C5 ⋊ C3⋊Dic3
HallC32×C11⋊C5 ⋊ Dic19
HallC11×C19 ⋊ C5×C3⋊Dic3
HallC32×C19 ⋊ Dic11⋊C5
HallC32×C11 ⋊ C5×Dic19
HallC11⋊C5 ⋊ C3⋊Dic57
SylowC19 ⋊ C3⋊(C3⋊(Dic11⋊C5))
SylowC11 ⋊ C5×C3⋊Dic57
SylowC32 ⋊ C19⋊(Dic11⋊C5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C3⋊(C3⋊(C19⋊AGL(1,𝔽11)))
commutator, hypocenterC32×C11×C19C4×C5
Fitting, socle, cosocleC2×C32×C11×C19C2×C5
prime radicalC2×C32×C19×C11⋊C5C2
LCSG ▷ C32×C11×C19
UCSC1 ◁ C2

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 7524

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2,5,3^2,11,19,2|a^2=g,c^a=c^2,d^a=d^2,e^a=e^10,f^a=f^18,e^b=e^4

Conjugacy classes & character table

1212 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37620, 167);