Finite Group Databasebeta

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C7×C3⋊(C3⋊#10596815) #49141640

order36540
automorphism?
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C3

Aliases

direct productC7×C3⋊(C3⋊#10596815)
unambiguous semidirect productC3⋊(C7×C3⋊#10596815)

Splittings

Fitting, socle, HallC32×C5×C7×C29 ⋊ C4
derived, HallC32×C5×C29 ⋊ C4×C7
HallC32×C7×C29 ⋊ AGL(1,𝔽5)
HallC5×C7×C29 ⋊ C3⋊Dic3
HallC32×C5×C7 ⋊ #1181
HallC32×C29 ⋊ C7×AGL(1,𝔽5)
HallC7×C29 ⋊ C3⋊ΓL(1,𝔽16)
HallC5×C29 ⋊ C7×C3⋊Dic3
HallC32×C7 ⋊ #10596815
HallC32×C5 ⋊ C7×#1181
HallC5×C7 ⋊ C3⋊(C3⋊#1181)
SylowC29 ⋊ C7×C3⋊ΓL(1,𝔽16)
SylowC32 ⋊ C7×#10596815
SylowC5 ⋊ C7×C3⋊(C3⋊#1181)

Subgroups

subgroupG/H
center, hypercenterC7C3⋊(C3⋊#10596815)
commutator, hypocenterC32×C5×C29C4×C7
FrattiniC1G
Fitting, socleC32×C5×C7×C29C4
prime radicalC32×C7×D145C2
cosocleC32×D145C2×C7
LCSG ▷ C32×C5×C29
UCSC1 ◁ C7

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:

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

Conjugacy classes & character table

2352 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 482);