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Cayley diagram

D115 #6593

order230
isoclinismisoclinic class of D115
automorphismAut(G) = AGL(1,𝔽5)×AGL(1,𝔽23) (order 10120) · Inn(G) = G · Out(G) = C4×C11
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD115
unambiguous semidirect productC5⋊D23
unambiguous semidirect productC23⋊D5

Splittings

Fitting, socle, derived, HallC5×C23 ⋊ C2
SylowC23 ⋊ D5
SylowC5 ⋊ D23

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C23C2
LCSG ▷ C5×C23
UCSC1

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

Polycyclic presentation:

2,5,23|b^a=b^4,c^a=c^22

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×D23, C2×D115, C3×D115, …
maximal normal subgroup ofD5×D23, C2×D115, C3×D115, …
maximal quotient ofDic115, C2×D115, C3×D115, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(230, 3);