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Cayley diagram

C4×C31⋊C3 #67309

order372
isoclinismisoclinic class of C31⋊C3
automorphismAut(G) = C2×AGL(1,𝔽31) (order 1860) · Inn(G) = C31⋊C3 · Out(G) = C22×C5
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC4×C31⋊C3
unambiguous semidirect product(C4×C31)⋊C3

Splittings

Fitting, HallC4×C31 ⋊ C3
derived, SylowC31 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC4C31⋊C3
commutator, hypocenterC31C3×C4
FrattiniC2C2×C31⋊C3
FittingC4×C31C3
socle, cosocleC2×C31C2×C3
prime radicalC2×C31⋊C3C2
LCSG ▷ C31
UCSC1 ◁ C4

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 124

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C31⋊C3)⋊C8, C8×C31⋊C3, (C31⋊C3)⋊Q8, …
maximal normal subgroup of(C31⋊C3)⋊C8, C8×C31⋊C3, (C31⋊C3)⋊Q8, …
maximal quotient ofC8×C31⋊C3, C2×C4×C31⋊C3, C4×C31⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(372, 2);