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Cayley diagram

D43⋊C3 #63106

order258
isoclinismisoclinic class of D43⋊C3
automorphismAut(G) = AGL(1,𝔽43) (order 1806) · Inn(G) = G · Out(G) = C7
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productD43⋊C3
unambiguous semidirect product(C43⋊C3)⋊C2

Splittings

Fitting, socle, derived, SylowC43 ⋊ C2×C3
HallC43⋊C3 ⋊ C2
HallD43 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC43C2×C3
LCSG ▷ C43
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 43

A minimal faithful permutation representation — 2 generators on 43 points:

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

Polycyclic presentation:

2,3,43|c^a=c^42,c^b=c^36

Conjugacy classes & character table

13 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D43⋊C3, C3×D43⋊C3, C3⋊(D43⋊C3), …
maximal normal subgroup ofC2×D43⋊C3, C3×D43⋊C3, C5×D43⋊C3, …
maximal quotient of(C43⋊C3)⋊C4, C2×D43⋊C3, D43⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(258, 1);