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C17×(C11⋊D11)⋊C3 #27134656

order12342
automorphism?
homologyAbelianization (H1) = C2×C3×C17 · Schur multiplier (H2) = C11

Aliases

direct productC17×(C11⋊D11)⋊C3
unambiguous semidirect product(C11⋊D11)⋊(C3×C17)
unambiguous semidirect product(C17×C11⋊D11)⋊C3

Splittings

Fitting, socle, HallC112×C17 ⋊ C2×C3
derived, SylowC112 ⋊ C2×C3×C17
HallC17×C112⋊C3 ⋊ C2
HallC17×C11⋊D11 ⋊ C3
HallC112⋊C3 ⋊ C2×C17
HallC11⋊D11 ⋊ C3×C17

Subgroups

subgroupG/H
center, hypercenterC17(C11⋊D11)⋊C3
commutator, cosocle, hypocenterC112C2×C3×C17
FrattiniC1G
Fitting, socleC112×C17C2×C3
LCSG ▷ C112
UCSC1 ◁ C17

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 50
transitive degreeμt(G) = 561

A minimal faithful permutation representation — 2 generators on 50 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC17×#10624391⋊C2, ((C11⋊D11)⋊C3)⋊D17, C17×#13089721, …
maximal normal subgroup ofC17×#10624391⋊C2, ((C11⋊D11)⋊C3)⋊D17, C17×#13089721, …
maximal quotient ofC17×(C11⋊Dic11)⋊C3, C2×C17×(C11⋊D11)⋊C3, C17×(C11⋊D11)⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12342, 12);