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C7×C3⋊D555 #37552147

order23310
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C3

Aliases

direct productC7×C3⋊D555
unambiguous semidirect productC3⋊(C7×D555)
unambiguous semidirect productC5⋊(C7×C3⋊D111)
unambiguous semidirect productC37⋊(C7×C3⋊D15)

Splittings

Fitting, socle, HallC32×C5×C7×C37 ⋊ C2
derived, HallC32×C5×C37 ⋊ C2×C7
HallC32×C7×C37 ⋊ D5
HallC5×C7×C37 ⋊ C3⋊D3
HallC32×C37 ⋊ C7×D5
HallC32×C5×C7 ⋊ D37
HallC7×C37 ⋊ C3⋊D15
HallC5×C37 ⋊ C7×C3⋊D3
HallC32×C7 ⋊ D185
HallC32×C5 ⋊ C7×D37
SylowC37 ⋊ C7×C3⋊D15
HallC5×C7 ⋊ C3⋊D111
SylowC32 ⋊ C7×D185
SylowC5 ⋊ C7×C3⋊D111

Subgroups

subgroupG/H
center, hypercenterC7C3⋊D555
commutator, cosocle, hypocenterC32×C5×C37C2×C7
FrattiniC1G
Fitting, socleC32×C5×C7×C37C2
LCSG ▷ C32×C5×C37
UCSC1 ◁ C7

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 55
transitive degreeμt(G) = 11655

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7)(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)(46,47)(48,49)(51,52)(54,55);(8,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,44,42,40,38,36,34,32,30,28,26,24,22,20,18,16,14,12,10)(45,46,48,49,47)(53,54,55);(50,51,52)

Conjugacy classes & character table

5838 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23310, 166);