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C5×C3⋊(C7⋊D161) #46798982

order33810
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C7

Aliases

direct productC5×C3⋊(C7⋊D161)
unambiguous semidirect productC3⋊(C5×C7⋊D161)
unambiguous semidirect productC7⋊(C5×D483)
unambiguous semidirect productC23⋊(C5×C3⋊(C7⋊D7))

Splittings

Fitting, socle, HallC3×C5×C72×C23 ⋊ C2
derived, HallC3×C72×C23 ⋊ C2×C5
HallC5×C72×C23 ⋊ D3
HallC72×C23 ⋊ C5×D3
HallC3×C5×C72 ⋊ D23
HallC3×C5×C23 ⋊ C7⋊D7
HallC5×C72 ⋊ D69
HallC3×C72 ⋊ C5×D23
HallC5×C23 ⋊ C3⋊(C7⋊D7)
HallC3×C23 ⋊ C5×C7⋊D7
SylowC72 ⋊ C5×D69
SylowC23 ⋊ C5×C3⋊(C7⋊D7)
HallC3×C5 ⋊ C7⋊D161
SylowC3 ⋊ C5×C7⋊D161

Subgroups

subgroupG/H
center, hypercenterC5C3⋊(C7⋊D161)
commutator, cosocle, hypocenterC3×C72×C23C2×C5
FrattiniC1G
Fitting, socleC3×C5×C72×C23C2
LCSG ▷ C3×C72×C23
UCSC1 ◁ C5

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 16905

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

8460 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33810, 86);