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C29⋊((C5⋊D5)⋊C9) #27189393

order13050
automorphism?
homologyAbelianization (H1) = C2×C9 · Schur multiplier (H2) = C5

Aliases

unambiguous semidirect productC29⋊((C5⋊D5)⋊C9)
unambiguous semidirect product(C5⋊D145)⋊C9

Splittings

derived, HallC52×C29 ⋊ C2×C9
HallC29×C52⋊C9 ⋊ C2
HallC5⋊D145 ⋊ C9
HallC52⋊C9 ⋊ D29
SylowC29 ⋊ (C5⋊D5)⋊C9
SylowC52 ⋊ C9×D29

Subgroups

subgroupG/H
center, Frattini, hypercenterC3C29⋊((C5⋊D5)⋊C3)
commutator, hypocenterC52×C29C2×C9
Fitting, socle, cosocleC3×C52×C29C2×C3
prime radicalC3×C5⋊D145C3
LCSG ▷ C52×C29
UCSC1 ◁ C3

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 1305

A minimal faithful permutation representation — 2 generators on 53 points:

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

Polycyclic presentation:

2,3,5^2,29,3|b^3=f,c^a=c^4,d^a=d^4,e^a=e^28,c^b=cd,d^b=c^2d^3

Conjugacy classes & character table

492 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#90646⋊D29, #15668526⋊C9, #15668525⋊C9, …
maximal normal subgroup of#90646⋊D29, #15668526⋊C9, #15668525⋊C9, …
maximal quotient of#15668515⋊C9, C2×C29⋊((C5⋊D5)⋊C9)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13050, 25);