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Cayley diagram

D31⋊C5 #64578

order310
isoclinismisoclinic class of D31⋊C5
automorphismAut(G) = AGL(1,𝔽31) (order 930) · Inn(G) = G · Out(G) = C3
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productD31⋊C5
unambiguous semidirect productΓL(1,𝔽32)⋊C2

Splittings

Fitting, socle, derived, SylowC31 ⋊ C2×C5
HallΓL(1,𝔽32) ⋊ C2
HallD31 ⋊ C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC31C2×C5
LCSG ▷ C31
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 31

A minimal faithful permutation representation — 2 generators on 31 points:

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

Polycyclic presentation:

2,5,31|c^a=c^30,c^b=c^16

Conjugacy classes & character table

13 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D31⋊C5, AGL(1,𝔽31), C3×D31⋊C5, …
maximal normal subgroup ofC2×D31⋊C5, AGL(1,𝔽31), C3×D31⋊C5, …
maximal quotient ofDic31⋊C5, C2×D31⋊C5, C3×D31⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(310, 1);