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Dic345 #13063404

order1380
isoclinismisoclinic class of D345
automorphismAut(G) order 121440
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic345
unambiguous semidirect productC3⋊Dic115
unambiguous semidirect productC23⋊Dic15

Splittings

derived, HallC3×C5×C23 ⋊ C4
HallC5×C23 ⋊ Dic3
HallC3×C23 ⋊ Dic5
SylowC23 ⋊ Dic15
HallC3×C5 ⋊ Dic23
SylowC5 ⋊ Dic69
SylowC3 ⋊ Dic115

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D345
commutator, hypocenterC3×C5×C23C4
Fitting, socle, prime radical, cosocleC2×C3×C5×C23C2
LCSG ▷ C3×C5×C23
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 1380

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

Polycyclic presentation:

2^2,3,5,23|a^2=b,c^a=c^2,d^a=d^4,e^a=e^22

Conjugacy classes & character table

348 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic1035, C3×Dic345, C3⋊Dic345, …
maximal normal subgroup ofC3×Dic345, C11×Dic345, C3⋊((C23⋊C11)⋊Dic5), …
maximal quotient ofDic1035, C3×Dic345, C3⋊Dic345, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1380, 7);