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Cayley diagram

C5×D23 #6592

order230
isoclinismisoclinic class of D23
automorphismAut(G) = C4×AGL(1,𝔽23) (order 2024) · Inn(G) = D23 · Out(G) = C4×C11
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D23
unambiguous semidirect productC23⋊(C2×C5)

Splittings

Fitting, socle, HallC5×C23 ⋊ C2
derived, SylowC23 ⋊ C2×C5

Subgroups

subgroupG/H
center, hypercenterC5D23
commutator, cosocle, hypocenterC23C2×C5
FrattiniC1G
Fitting, socleC5×C23C2
LCSG ▷ C23
UCSC1 ◁ C5

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 115

A minimal faithful permutation representation — 2 generators on 28 points:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×D23, C2×C5×D23, C3×C5×D23, …
maximal normal subgroup ofD5×D23, C2×C5×D23, C3×C5×D23, …
maximal quotient ofC5×Dic23, C2×C5×D23, C3×C5×D23, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(230, 2);