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C5⋊(C11⋊(D109⋊C3)) #49112593

order35970
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(C11⋊(D109⋊C3))
unambiguous semidirect productC11⋊(C5⋊(D109⋊C3))
unambiguous semidirect product(C109⋊C3)⋊D55
unambiguous semidirect productD5995⋊C3

Splittings

Fitting, socle, derived, HallC5×C11×C109 ⋊ C2×C3
HallC5×C11×C109⋊C3 ⋊ C2
HallD5995 ⋊ C3
HallC11×C109⋊C3 ⋊ D5
HallC5×C109⋊C3 ⋊ D11
HallC11×C109 ⋊ C3×D5
HallC5×C109 ⋊ C3×D11
HallC109⋊C3 ⋊ D55
SylowC109 ⋊ C3×D55
HallC5×C11 ⋊ D109⋊C3
SylowC11 ⋊ C5⋊(D109⋊C3)
SylowC5 ⋊ C11⋊(D109⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C11×C109C2×C3
LCSG ▷ C5×C11×C109
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 125
transitive degreeμt(G) = 5995

A minimal faithful permutation representation — 2 generators on 125 points:

(2,3)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(18,22,23,19,20,24)(21,30,31,25,26,40)(27,43,44,33,34,57)(28,42,38,36,32,29)(35,62,63,46,47,80)(37,60,61,49,45,54)(39,41,59,51,52,55)(48,85,86,65,66,107)(50,83,84,68,64,77)(53,58,82,70,71,78)(56,96,94,74,75,73)(67,112,113,88,89,125)(69,110,111,90,87,104)(72,81,109,91,92,105)(76,117,118,97,98,100)(79,121,115,101,102,95)(93,108,119,114,99,124)(103,123,116,106,122,120);(1,2,4,5,3)(6,7,9,11,13,15,16,14,12,10,8)(17,18,21,27,35,48,67,86,109,121,117,110,85,108,97,74,52,26,32,40,54,73,95,116,118,84,44,59,60,43,58,79,103,92,47,64,75,78,80,104,114,91,70,51,38,23,22,28,37,50,69,89,113,119,122,123,124,125,112,90,68,49,36,20,24,29,39,53,72,93,111,63,82,96,83,62,81,106,101,71,34,45,55,57,77,100,120,115,94,61,31,42,30,41,56,76,99,66,87,98,102,105,107,88,65,46,33,25,19)

Polycyclic presentation:

2,3,5,11,109|c^a=c^4,d^a=d^10,e^a=e^108,e^b=e^63

Conjugacy classes & character table

1077 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35970, 33);