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C5⋊(C23⋊(D49⋊C3)) #46798920

order33810
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(C23⋊(D49⋊C3))
unambiguous semidirect productC23⋊(C5⋊(D49⋊C3))
unambiguous semidirect product(C49⋊C3)⋊D115
unambiguous semidirect productD5635⋊C3

Splittings

Fitting, derived, HallC5×C23×C49 ⋊ C2×C3
HallC5×C23×C49⋊C3 ⋊ C2
HallD5635 ⋊ C3
HallC23×C49⋊C3 ⋊ D5
HallC23×C49 ⋊ C3×D5
HallC5×C49⋊C3 ⋊ D23
HallC5×C49 ⋊ C3×D23
HallC49⋊C3 ⋊ D115
HallC5×C23 ⋊ D49⋊C3
SylowC49 ⋊ C3×D115
SylowC23 ⋊ C5⋊(D49⋊C3)
SylowC5 ⋊ C23⋊(D49⋊C3)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC5×C23×C49C2×C3
FrattiniC7C5⋊(ΓL(1,𝔽8)⋊D23)
socleC5×C7×C23AGL(1,𝔽7)
LCSG ▷ C5×C23×C49
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 77
transitive degreeμt(G) = 5635

A minimal faithful permutation representation — 2 generators on 77 points:

(2,3)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(30,38,40,32,33,41)(31,50,52,36,37,53)(34,62,49,43,44,71)(35,67,68,47,48,77)(39,46,45,51,56,64)(42,65,74,60,61,75)(54,66,69,72,55,58)(57,70,59,73,63,76);(1,2,4,5,3)(6,7,9,11,13,15,17,19,21,23,25,27,28,26,24,22,20,18,16,14,12,10,8)(29,74,46,69,68,55,57,36,72,34,59,64,38,39,50,49,70,41,61,60,32,52,35,48,62,44,67,47,53,30,42,65,40,63,71,37,51,33,45,76,43,54,31,73,66,77,58,56,75)

Polycyclic presentation:

2,3,5,7,23,7|d^7=f^6,c^a=c^4,d^a=d^6f,e^a=e^22,f^a=f^6,d^b=d^2f^3,f^b=f^2

Conjugacy classes & character table

1097 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33810, 24);