Finite Group Databasebeta

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Cayley diagram

D184 #67264

order368
isoclinismisoclinic class of #67263
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD184

Splittings

FittingC8×C23 ⋊ C2
SylowC23 ⋊ D8

Subgroups

subgroupG/H
centerC2D92
commutator, cosocleC4×C23C22
FrattiniC4C2×D23
FittingC8×C23C2
socleC2×C23D4
hypercenterC8D23
hypocenterC23D8
LCSG ▷ C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C4 ◁ C8

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 184

A minimal faithful permutation representation — 2 generators on 31 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);(2,8)(3,7)(4,6)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)

Polycyclic presentation:

2^4,23|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^22

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD368, #10624982, #10625008, …
maximal normal subgroup ofD368, #10624982, #10625008, …
maximal quotient ofD368, #10624982, Dic184, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(368, 6);