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Cayley diagram

C2×C43⋊C3 #63107

order258
isoclinismisoclinic class of C43⋊C3
automorphismAut(G) = AGL(1,𝔽43) (order 1806) · Inn(G) = C43⋊C3 · Out(G) = C2×C7
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC2×C43⋊C3
unambiguous semidirect product(C2×C43)⋊C3

Splittings

Fitting, socle, HallC2×C43 ⋊ C3
derived, SylowC43 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2C43⋊C3
commutator, cosocle, hypocenterC43C2×C3
FrattiniC1G
Fitting, socleC2×C43C3
LCSG ▷ C43
UCSC1 ◁ C2

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 86

A minimal faithful permutation representation — 2 generators on 45 points:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C43⋊C3)⋊C4, C4×C43⋊C3, C2×D43⋊C3, …
maximal normal subgroup of(C43⋊C3)⋊C4, C4×C43⋊C3, C2×D43⋊C3, …
maximal quotient ofC4×C43⋊C3, C22×C43⋊C3, C2×C43⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(258, 2);