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Cayley diagram

Dic54 #6167

order216
isoclinismisoclinic class of Dic54
automorphismAut(G) = D4×AGL(1,ℤ/27) (order 3888) · Inn(G) = C2×D27 · Out(G) = C22×C9
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic54
unambiguous semidirect productC27⋊Q8

Splittings

SylowC27 ⋊ Q8

Subgroups

subgroupG/H
centerC2C2×D27
commutator, cosocleC2×C27C22
FrattiniC2×C9C2×D3
FittingC4×C27C2
socle, prime radicalC2×C3C2×D9
hypercenterC4D27
hypocenterC27Q8
LCSG ▷ C2×C27 ▷ C27
UCSC1 ◁ C2 ◁ C4

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 216

A minimal faithful permutation representation — 2 generators on 35 points:

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

Polycyclic presentation:

2^3,3^3|a^2=c,b^2=c,d^3=e^2f,e^3=f^2,b^a=bc,d^a=d^2e,e^a=e^2f,f^a=f^2

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic108, #88360, #88368, …
maximal normal subgroup ofDic108, #88360, #88368, …
maximal quotient ofDic27⋊C4, C4⋊Dic27, C2×Dic54, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 4);