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Dic15400 #72293479

order61600
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic15400

Splittings

HallC7×C11×C25 ⋊ Q32
HallC11×C25 ⋊ Dic56
HallC7×C25 ⋊ Dic88
HallC7×C11 ⋊ Dic200
SylowC25 ⋊ Dic616
SylowC11 ⋊ Dic1400
SylowC7 ⋊ Dic2200

Subgroups

subgroupG/H
centerC2D15400
commutator, cosocleC7×C8×C11×C25C22
FrattiniC5×C8C2×D385
FittingC7×C11×C16×C25C2
socle, prime radicalC2×C5×C7×C11D40
hypercenterC16D1925
hypocenterC7×C11×C25Q32
LCSG ▷ C7×C8×C11×C25 ▷ C4×C7×C11×C25 ▷ C2×C7×C11×C25 ▷ C7×C11×C25
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

144 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 75
transitive degreeμt(G) = 61600

A minimal faithful permutation representation — 2 generators on 75 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(59,64)(60,63)(61,62)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2^5,5,7,11,5|a^2=e,b^2=cd,c^2=de,d^2=e,f^5=i,b^a=bc,c^a=cd,d^a=de,f^a=f^4i^4,g^a=g^6,h^a=h^10,i^a=i^4

Conjugacy classes & character table

15403 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 7, 11, 5]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^4*F.9^4);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^10);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);