Finite Group Databasebeta

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C5×C3⋊(C3⋊(C7⋊#1181)) #49141632

order36540
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊(C3⋊(C7⋊#1181))
unambiguous semidirect productC3⋊(C5×C3⋊(C7⋊#1181))
unambiguous semidirect productC7⋊(C5×C3⋊(C3⋊#1181))

Splittings

Fitting, socle, HallC32×C5×C7×C29 ⋊ C4
derived, HallC32×C7×C29 ⋊ C4×C5
HallC32×C5×C29 ⋊ Dic7
HallC5×C7×C29 ⋊ C3⋊Dic3
HallC32×C5×C7 ⋊ #1181
HallC32×C29 ⋊ C5×Dic7
HallC7×C29 ⋊ C5×C3⋊Dic3
HallC5×C29 ⋊ C3⋊Dic21
HallC32×C7 ⋊ C5×#1181
HallC32×C5 ⋊ C7⋊#1181
HallC5×C7 ⋊ C3⋊(C3⋊#1181)
SylowC29 ⋊ C5×C3⋊Dic21
SylowC32 ⋊ C5×C7⋊#1181
SylowC7 ⋊ C5×C3⋊(C3⋊#1181)

Subgroups

subgroupG/H
center, hypercenterC5C3⋊(C3⋊(C7⋊#1181))
commutator, hypocenterC32×C7×C29C4×C5
FrattiniC1G
Fitting, socleC32×C5×C7×C29C4
prime radicalC32×C5×C7×D29C2
cosocleC32×C7×D29C2×C5
LCSG ▷ C32×C7×C29
UCSC1 ◁ C5

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 47
transitive degreeμt(G) = 9135

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5)(7,8)(10,11)(13,14)(15,16)(17,18)(20,21,22,24)(23,26,27,33)(25,30,31,29)(28,35,36,44)(32,37,39,41)(34,42,43,38)(40,45,46,47);(9,10,11)(12,13,15,17,18,16,14)(19,20,23,28,37,26,34,44,47,46,39,31,24,29,38,42,30,21,25,32,40,45,35,43,33,41,36,27,22);(6,7,8)

Conjugacy classes & character table

2535 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 474);