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Dic975 #16839004

order3900
automorphismAut(G) order 936000
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic975
unambiguous semidirect productC3⋊Dic325

Splittings

derived, HallC3×C13×C25 ⋊ C4
HallC13×C25 ⋊ Dic3
HallC3×C25 ⋊ Dic13
HallC3×C13 ⋊ Dic25
SylowC25 ⋊ Dic39
SylowC13 ⋊ Dic75
SylowC3 ⋊ Dic325

Subgroups

subgroupG/H
center, hypercenterC2D975
commutator, hypocenterC3×C13×C25C4
FrattiniC2×C5D195
Fitting, cosocleC2×C3×C13×C25C2
socle, prime radicalC2×C3×C5×C13D5
LCSG ▷ C3×C13×C25
UCSC1 ◁ C2

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 3900

A minimal faithful permutation representation — 2 generators on 45 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)(33,34,35,36,37,38,39,40,41,42,43,44,45);(1,3,2,4)(6,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)

Polycyclic presentation:

2,3,5,13,2,5|a^2=e,c^5=f^4,b^a=b^2,c^a=c^4f,d^a=d^12,f^a=f^4

Conjugacy classes & character table

978 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic2925, C3⋊(Dic325⋊C3), C3×Dic975, …
maximal normal subgroup ofC3⋊(Dic325⋊C3), C3×Dic975, C7×Dic975, …
maximal quotient ofDic2925, C3×Dic975, C3⋊Dic975, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3900, 12);