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Cayley diagram

C2×Dic31 #6938

order248
isoclinismisoclinic class of D31
automorphismAut(G) = D4×AGL(1,𝔽31) (order 7440) · Inn(G) = D31 · Out(G) = C3×C5×D4
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×Dic31
unambiguous semidirect product(C2×C31)⋊C4

Splittings

derived, SylowC31 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC22D31
commutator, hypocenterC31C2×C4
FrattiniC2C2×D31
Fitting, socle, prime radicalC22×C31C2
cosocleC2×C31C22
LCSG ▷ C31
UCSC1 ◁ C22

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 248

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×Dic31, Dic31⋊C4, C4⋊Dic31, …
maximal normal subgroup ofC4×Dic31, Dic31⋊C4, C4⋊Dic31, …
maximal quotient ofC2×C31⋊C8, #92474, C4×Dic31, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(248, 6);