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Cayley diagram

C22⋊D31 #6939

order248
isoclinismisoclinic class of Dic62
automorphismAut(G) = C22×AGL(1,𝔽31) (order 3720) · Inn(G) = C2×D31 · Out(G) = C2×C3×C5
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC22⋊D31

Splittings

FittingC22×C31 ⋊ C2
SylowC31 ⋊ D4

Subgroups

subgroupG/H
center, FrattiniC2C2×D31
commutator, socle, cosocleC2×C31C22
FittingC22×C31C2
hypercenterC22D31
hypocenterC31D4
LCSG ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#92495, D4×D31, #92497, …
maximal normal subgroup of#92495, D4×D31, #92497, …
maximal quotient ofDic31⋊C4, (C2×D31)⋊C4, #92479, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(248, 7);