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Dic525 #14523532

order2100
isoclinismisoclinic class of D525
automorphismAut(G) order 252000
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic525
unambiguous semidirect productC3⋊Dic175
unambiguous semidirect productC7⋊Dic75

Splittings

derived, HallC3×C7×C25 ⋊ C4
HallC7×C25 ⋊ Dic3
HallC3×C25 ⋊ Dic7
SylowC25 ⋊ Dic21
HallC3×C7 ⋊ Dic25
SylowC7 ⋊ Dic75
SylowC3 ⋊ Dic175

Subgroups

subgroupG/H
center, hypercenterC2D525
commutator, hypocenterC3×C7×C25C4
FrattiniC2×C5D105
Fitting, cosocleC2×C3×C7×C25C2
socle, prime radicalC2×C3×C5×C7D5
LCSG ▷ C3×C7×C25
UCSC1 ◁ C2

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 39
transitive degreeμt(G) = 2100

A minimal faithful permutation representation — 2 generators on 39 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);(1,3,2,4)(6,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(34,39)(35,38)(36,37)

Polycyclic presentation:

2,3,5,7,2,5|a^2=e,c^5=f^4,b^a=b^2,c^a=c^4f,d^a=d^6,f^a=f^4

Conjugacy classes & character table

528 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3675, C7×Dic525, C3⋊(C7⋊Dic175), …
maximal normal subgroup ofC7×Dic525, C11×Dic525, C13×Dic525, …
maximal quotient ofDic3675, C7×Dic525, C3⋊(C7⋊Dic175), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 12);