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Cayley diagram

Dic50 #6017

order200
isoclinismisoclinic class of Dic50
automorphismAut(G) = D4×AGL(1,ℤ/25) (order 4000) · Inn(G) = C2×D25 · Out(G) = C2×C4×C5
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic50
unambiguous semidirect productC25⋊Q8

Splittings

SylowC25 ⋊ Q8

Subgroups

subgroupG/H
centerC2C2×D25
commutator, cosocleC2×C25C22
Frattini, socle, prime radicalC2×C5C2×D5
FittingC4×C25C2
hypercenterC4D25
hypocenterC25Q8
LCSG ▷ C2×C25 ▷ C25
UCSC1 ◁ C2 ◁ C4

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 200

A minimal faithful permutation representation — 2 generators on 33 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic100, #87713, #87721, …
maximal normal subgroup ofDic100, #87713, #87721, …
maximal quotient of#87718, C4⋊Dic25, C2×Dic50, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 4);