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Dic215 #11720020

order860
isoclinismisoclinic class of D215
automorphismAut(G) order 72240
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic215
unambiguous semidirect productC43⋊Dic5

Splittings

derived, HallC5×C43 ⋊ C4
SylowC43 ⋊ Dic5
SylowC5 ⋊ Dic43

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D215
commutator, hypocenterC5×C43C4
Fitting, socle, prime radical, cosocleC2×C5×C43C2
LCSG ▷ C5×C43
UCSC1 ◁ C2

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 52
transitive degreeμt(G) = 860

A minimal faithful permutation representation — 2 generators on 52 points:

(1,2)(3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52);(1,3,2,4)(6,9)(7,8)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)

Polycyclic presentation:

2^2,5,43|a^2=b,c^a=c^4,d^a=d^42

Conjugacy classes & character table

218 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic5×D43, D5×Dic43, #13133884, …
maximal normal subgroup ofDic5×D43, D5×Dic43, #13133884, …
maximal quotient ofC43⋊#168, C2×Dic215, C3×Dic215, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(860, 3);