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Dic385 #13095356

order1540
isoclinismisoclinic class of D385
automorphismAut(G) order 184800
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic385
unambiguous semidirect productC7⋊Dic55
unambiguous semidirect productC11⋊Dic35

Splittings

derived, HallC5×C7×C11 ⋊ C4
HallC7×C11 ⋊ Dic5
HallC5×C11 ⋊ Dic7
HallC5×C7 ⋊ Dic11
SylowC11 ⋊ Dic35
SylowC7 ⋊ Dic55
SylowC5 ⋊ Dic77

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D385
commutator, hypocenterC5×C7×C11C4
Fitting, socle, prime radical, cosocleC2×C5×C7×C11C2
LCSG ▷ C5×C7×C11
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 1540

A minimal faithful permutation representation — 2 generators on 27 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);(1,3,2,4)(6,9)(7,8)(11,16)(12,15)(13,14)(18,27)(19,26)(20,25)(21,24)(22,23)

Polycyclic presentation:

2^2,5,7,11|a^2=b,c^a=c^4,d^a=d^6,e^a=e^10

Conjugacy classes & character table

388 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×Dic385, Dic1155, C11⋊(ΓL(1,𝔽8)⋊Dic5), …
maximal normal subgroup ofC3×Dic385, C11⋊(ΓL(1,𝔽8)⋊Dic5), C11×Dic385, …
maximal quotient ofC3×Dic385, Dic1155, C7⋊(C11⋊#4379), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1540, 11);