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Dic285 #11765883

order1140
isoclinismisoclinic class of D285
automorphismAut(G) order 82080
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic285
unambiguous semidirect productC3⋊Dic95
unambiguous semidirect productC19⋊Dic15

Splittings

derived, HallC3×C5×C19 ⋊ C4
HallC5×C19 ⋊ Dic3
HallC3×C19 ⋊ Dic5
SylowC19 ⋊ Dic15
HallC3×C5 ⋊ Dic19
SylowC5 ⋊ Dic57
SylowC3 ⋊ Dic95

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D285
commutator, hypocenterC3×C5×C19C4
Fitting, socle, prime radical, cosocleC2×C3×C5×C19C2
LCSG ▷ C3×C5×C19
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 1140

A minimal faithful permutation representation — 2 generators on 31 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);(1,3,2,4)(6,7)(9,12)(10,11)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)

Polycyclic presentation:

2^2,3,5,19|a^2=b,c^a=c^2,d^a=d^4,e^a=e^18

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic855, C3⋊((C19⋊C3)⋊Dic5), C3×Dic285, …
maximal normal subgroup ofC3⋊((C19⋊C3)⋊Dic5), C3×Dic285, C11×Dic285, …
maximal quotient ofDic855, C3×Dic285, C3⋊Dic285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1140, 11);