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Cayley diagram

C2×C31⋊C3 #4414

order186
isoclinismisoclinic class of C31⋊C3
automorphismAut(G) = AGL(1,𝔽31) (order 930) · Inn(G) = C31⋊C3 · Out(G) = C2×C5
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC2×C31⋊C3
unambiguous semidirect product(C2×C31)⋊C3

Splittings

Fitting, socle, HallC2×C31 ⋊ C3
derived, SylowC31 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2C31⋊C3
commutator, cosocle, hypocenterC31C2×C3
FrattiniC1G
Fitting, socleC2×C31C3
LCSG ▷ C31
UCSC1 ◁ C2

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 62

A minimal faithful permutation representation — 2 generators on 33 points:

(1,2)(4,5,7)(6,9,14)(8,12,11)(10,16,23)(13,19,21)(15,17,18)(20,26,30)(22,27,28)(24,31,25)(29,32,33);(3,4,6,10,17,7,11,18,25,31,14,21,28,16,24,23,30,33,26,9,15,22,29,32,19,5,8,13,20,27,12)

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C31⋊C3)⋊C4, C4×C31⋊C3, C2×D31⋊C3, …
maximal normal subgroup of(C31⋊C3)⋊C4, C4×C31⋊C3, C2×D31⋊C3, …
maximal quotient ofC4×C31⋊C3, C22×C31⋊C3, C2×C31⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(186, 2);