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Cayley diagram

D124 #6937

order248
isoclinismisoclinic class of Dic62
automorphismAut(G) = D4×AGL(1,𝔽31) (order 7440) · Inn(G) = C2×D31 · Out(G) = C22×C3×C5
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD124
unambiguous semidirect productC4⋊D31

Splittings

FittingC4×C31 ⋊ C2
SylowC31 ⋊ D4

Subgroups

subgroupG/H
center, FrattiniC2C2×D31
commutator, socle, cosocleC2×C31C22
FittingC4×C31C2
hypercenterC4D31
hypocenterC31D4
LCSG ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C4

16 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:

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

Polycyclic presentation:

2^3,31|b^2=c,b^a=bc,d^a=d^30

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#92470, D248, #92479, …
maximal normal subgroup of#92470, D248, #92479, …
maximal quotient of#92470, D248, Dic124, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(248, 5);