D₃₂₃₀₀ #72302020

order	64600
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₃₀₀

Splittings

Fitting	C₄×C₁₇×C₁₉×C₂₅ ⋊ C₂
Hall	C₁₇×C₁₉×C₂₅ ⋊ D₄
Hall	C₁₉×C₂₅ ⋊ D₆₈
Hall	C₁₇×C₂₅ ⋊ D₇₆
Hall	C₁₇×C₁₉ ⋊ D₁₀₀
Sylow	C₂₅ ⋊ D₁₂₉₂
Sylow	C₁₉ ⋊ D₁₇₀₀
Sylow	C₁₇ ⋊ D₁₉₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₈₀₇₅
commutator, cosocle	C₂×C₁₇×C₁₉×C₂₅	C₂²
Frattini	C₂×C₅	C₂×D₁₆₁₅
Fitting	C₄×C₁₇×C₁₉×C₂₅	C₂
socle	C₂×C₅×C₁₇×C₁₉	C₂×D₅
hypercenter	C₄	D₈₀₇₅
hypocenter	C₁₇×C₁₉×C₂₅	D₄
LCS	G ▷ C₂×C₁₇×C₁₉×C₂₅ ▷ C₁₇×C₁₉×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 32300

A minimal faithful permutation representation — 2 generators on 65 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);(2,4)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2^3,5,17,19,5|b^2=c,d^5=g,b^a=bc,d^a=d^4g^4,e^a=e^16,f^a=f^18,g^a=g^4

Conjugacy classes & character table

16153 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 17, 19, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.7^4);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32300 #72302020