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C7×D19×D69 #49144807

order36708
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×D19×D69
unambiguous semidirect product(C3×D19)⋊(C7×D23)
unambiguous semidirect product(C23×D19)⋊(D3×C7)

Splittings

Fitting, socle, HallC3×C7×C19×C23 ⋊ C22
derived, HallC3×C19×C23 ⋊ C22×C7
HallC7×C19×C23 ⋊ C2×D3
HallC3×C7×C23 ⋊ C2×D19
HallC19×C23 ⋊ C2×D3×C7
HallC3×C7×C19 ⋊ C2×D23
HallC7×C23 ⋊ D3×D19
HallC7×C19 ⋊ C2×D69
HallC3×C23 ⋊ C2×C7×D19
HallC3×C19 ⋊ C2×C7×D23
SylowC23 ⋊ D3×C7×D19
HallC3×C7 ⋊ D19×D23
SylowC19 ⋊ C2×C7×D69
SylowC3 ⋊ C7×D19×D23

Subgroups

subgroupG/H
center, hypercenterC7D19×D69
commutator, cosocle, hypocenterC3×C19×C23C22×C7
FrattiniC1G
Fitting, socleC3×C7×C19×C23C22
LCSG ▷ C3×C19×C23
UCSC1 ◁ C7

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 52
transitive degreeμt(G) = 9177

A minimal faithful permutation representation — 2 generators on 52 points:

(2,3)(4,5,6,7,8,9,10)(11,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52);(1,2,3)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31,33,35,37,39,41,43,45,47,49,51,52,50,48,46,44,42,40,38,36,34,32)

Conjugacy classes & character table

2772 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36708, 116);