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Cayley diagram

Dic46 #4402

order184
isoclinismisoclinic class of Dic46
automorphismAut(G) = D4×AGL(1,𝔽23) (order 4048) · Inn(G) = C2×D23 · Out(G) = C22×C11
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic46
unambiguous semidirect productC23⋊Q8

Splittings

SylowC23 ⋊ Q8

Subgroups

subgroupG/H
center, FrattiniC2C2×D23
commutator, socle, prime radical, cosocleC2×C23C22
FittingC4×C23C2
hypercenterC4D23
hypocenterC23Q8
LCSG ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C4

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

Polycyclic presentation:

2^3,23|a^2=c,b^2=c,b^a=bc,d^a=d^22

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#67263, Dic92, #67273, …
maximal normal subgroup of#67263, Dic92, #67273, …
maximal quotient ofDic23⋊C4, C4⋊Dic23, C2×Dic46, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(184, 3);