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C3×D6307 #49180913

order37842
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D6307
unambiguous semidirect productC17⋊(C3×D371)
unambiguous semidirect productC53⋊(C3×D119)

Splittings

Fitting, socle, HallC3×C7×C17×C53 ⋊ C2
derived, HallC7×C17×C53 ⋊ C2×C3
HallC3×C17×C53 ⋊ D7
HallC3×C7×C53 ⋊ D17
HallC17×C53 ⋊ C3×D7
HallC7×C53 ⋊ C3×D17
HallC3×C7×C17 ⋊ D53
HallC3×C53 ⋊ D119
HallC7×C17 ⋊ C3×D53
SylowC53 ⋊ C3×D119
HallC3×C17 ⋊ D371
HallC3×C7 ⋊ D901
SylowC17 ⋊ C3×D371
SylowC7 ⋊ C3×D901

Subgroups

subgroupG/H
center, hypercenterC3D6307
commutator, cosocle, hypocenterC7×C17×C53C2×C3
FrattiniC1G
Fitting, socleC3×C7×C17×C53C2
LCSG ▷ C7×C17×C53
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 80
transitive degreeμt(G) = 18921

A minimal faithful permutation representation — 2 generators on 80 points:

(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80);(5,10)(6,9)(7,8)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Conjugacy classes & character table

9465 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37842, 9);