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Cayley diagram

Dic23 #785

order92
isoclinismisoclinic class of D23
automorphismAut(G) = C2×AGL(1,𝔽23) (order 1012) · Inn(G) = D23 · Out(G) = C2×C11
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic23
unambiguous semidirect productC23⋊C4

Splittings

derived, SylowC23 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D23
commutator, hypocenterC23C4
Fitting, socle, prime radical, cosocleC2×C23C2
LCSG ▷ C23
UCSC1 ◁ C2

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 92

A minimal faithful permutation representation — 2 generators on 27 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);(1,3,2,4)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)

Polycyclic presentation:

2^2,23|a^2=b,c^a=c^22

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic46, C4×D23, C2×Dic23, …
maximal normal subgroup ofDic46, C4×D23, C2×Dic23, …
maximal quotient ofC23⋊C8, C2×Dic23, C3×Dic23, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(92, 1);