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C2×D3999 #30630630

order15996
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD7998
direct productC2×D3999
unambiguous semidirect product(C2×C3)⋊D1333
unambiguous semidirect product(C2×C31)⋊D129
unambiguous semidirect product(C2×C43)⋊D93

Splittings

Fitting, socleC2×C3×C31×C43 ⋊ C2
derived, HallC3×C31×C43 ⋊ C22
HallC31×C43 ⋊ C2×D3
HallC3×C43 ⋊ C2×D31
HallC3×C31 ⋊ C2×D43
SylowC43 ⋊ C2×D93
SylowC31 ⋊ C2×D129
SylowC3 ⋊ C2×D1333

Subgroups

subgroupG/H
center, hypercenterC2D3999
commutator, cosocle, hypocenterC3×C31×C43C22
FrattiniC1G
Fitting, socleC2×C3×C31×C43C2
LCSG ▷ C3×C31×C43
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 79
transitive degreeμt(G) = 7998

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

Conjugacy classes & character table

4002 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD15996
maximal normal subgroup ofD15996
maximal quotient ofDic7998, D15996

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15996, 58);