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Cayley diagram

Dic115 #90745

order460
isoclinismisoclinic class of D115
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic115
unambiguous semidirect productC23⋊Dic5

Splittings

derived, HallC5×C23 ⋊ C4
SylowC23 ⋊ Dic5
SylowC5 ⋊ Dic23

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D115
commutator, hypocenterC5×C23C4
Fitting, socle, prime radical, cosocleC2×C5×C23C2
LCSG ▷ C5×C23
UCSC1 ◁ C2

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 460

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2)(3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32);(1,3,2,4)(6,9)(7,8)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)

Polycyclic presentation:

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

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic5×D23, D5×Dic23, #11745065, …
maximal normal subgroup ofDic5×D23, D5×Dic23, #11745065, …
maximal quotient ofC23⋊#168, C2×Dic115, C3×Dic115, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(460, 3);