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C3⋊((C11⋊C5)⋊D41) #27214306

order13530
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊((C11⋊C5)⋊D41)
unambiguous semidirect product(C11⋊C5)⋊D123

Splittings

Fitting, socle, derived, HallC3×C11×C41 ⋊ C2×C5
HallC3×C41×C11⋊C5 ⋊ C2
HallD1353 ⋊ C5
HallC41×C11⋊C5 ⋊ D3
HallC11×C41 ⋊ C5×D3
HallC3×C11⋊C5 ⋊ D41
HallC3×C41 ⋊ AGL(1,𝔽11)
HallC11⋊C5 ⋊ D123
SylowC41 ⋊ C3⋊AGL(1,𝔽11)
HallC3×C11 ⋊ C5×D41
SylowC11 ⋊ C5×D123
SylowC3 ⋊ (C11⋊C5)⋊D41

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C11×C41C2×C5
LCSG ▷ C3×C11×C41
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 55
transitive degreeμt(G) = 1353

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

Polycyclic presentation:

2,3,5,11,41|b^a=b^2,d^a=d^10,e^a=e^40,d^c=d^4

Conjugacy classes & character table

438 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD41×C3⋊AGL(1,𝔽11), AGL(1,𝔽11)×D123, D3×(C11⋊C5)⋊D41, …
maximal normal subgroup ofD41×C3⋊AGL(1,𝔽11), AGL(1,𝔽11)×D123, D3×(C11⋊C5)⋊D41, …
maximal quotient ofC3⋊((C11⋊C5)⋊Dic41), C2×C3⋊((C11⋊C5)⋊D41)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13530, 60);