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C3×D3995 #37561790

order23970
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D3995
unambiguous semidirect productC5⋊(C3×D799)
unambiguous semidirect productC17⋊(C3×D235)
unambiguous semidirect productC47⋊(C3×D85)

Splittings

Fitting, socle, HallC3×C5×C17×C47 ⋊ C2
derived, HallC5×C17×C47 ⋊ C2×C3
HallC3×C17×C47 ⋊ D5
HallC17×C47 ⋊ C3×D5
HallC3×C5×C47 ⋊ D17
HallC3×C5×C17 ⋊ D47
HallC5×C47 ⋊ C3×D17
HallC3×C47 ⋊ D85
HallC5×C17 ⋊ C3×D47
HallC3×C17 ⋊ D235
SylowC47 ⋊ C3×D85
SylowC17 ⋊ C3×D235
HallC3×C5 ⋊ D799
SylowC5 ⋊ C3×D799

Subgroups

subgroupG/H
center, hypercenterC3D3995
commutator, cosocle, hypocenterC5×C17×C47C2×C3
FrattiniC1G
Fitting, socleC3×C5×C17×C47C2
LCSG ▷ C5×C17×C47
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 72
transitive degreeμt(G) = 11985

A minimal faithful permutation representation — 2 generators on 72 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);(5,8)(6,7)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Conjugacy classes & character table

5997 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23970, 8);