D₉₈₅₅ #72215001

order	19710
automorphism	Aut(G) order 51088320
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₉₈₅₅

Splittings

Fitting, derived, Hall	C₅×C₂₇×C₇₃ ⋊ C₂
Hall	C₂₇×C₇₃ ⋊ D₅
Hall	C₅×C₇₃ ⋊ D₂₇
Hall	C₅×C₂₇ ⋊ D₇₃
Sylow	C₇₃ ⋊ D₁₃₅
Sylow	C₂₇ ⋊ D₃₆₅
Sylow	C₅ ⋊ D₁₉₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₂₇×C₇₃	C₂
Frattini	C₉	D₁₀₉₅
socle	C₃×C₅×C₇₃	D₉
LCS	G ▷ C₅×C₂₇×C₇₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 105
transitive degree	μ_t(G) = 9855

A minimal faithful permutation representation — 2 generators on 105 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,32)(30,31)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2,3,5,73,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^4,d^a=d^72,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

4929 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₉₈₅₅, D₂₉₅₆₅
maximal normal subgroup of	C₂×D₉₈₅₅
maximal quotient of	Dic₉₈₅₅, C₂×D₉₈₅₅, D₂₉₅₆₅

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");;
gap> coll := SingleCollector(F, [2, 3, 5, 73, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^72);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9855 #72215001