D₃₀₈₂₀ #72296547

order	61640
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₈₂₀

Splittings

Fitting	C₄×C₅×C₂₃×C₆₇ ⋊ C₂
Hall	C₅×C₂₃×C₆₇ ⋊ D₄
Hall	C₂₃×C₆₇ ⋊ D₂₀
Hall	C₅×C₆₇ ⋊ D₉₂
Hall	C₅×C₂₃ ⋊ D₂₆₈
Sylow	C₆₇ ⋊ D₄₆₀
Sylow	C₂₃ ⋊ D₁₃₄₀
Sylow	C₅ ⋊ D₆₁₆₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₇₀₅
commutator, socle, cosocle	C₂×C₅×C₂₃×C₆₇	C₂²
Fitting	C₄×C₅×C₂₃×C₆₇	C₂
hypercenter	C₄	D₇₇₀₅
hypocenter	C₅×C₂₃×C₆₇	D₄
LCS	G ▷ C₂×C₅×C₂₃×C₆₇ ▷ C₅×C₂₃×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 30820

A minimal faithful permutation representation — 2 generators on 99 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);(2,4)(6,9)(7,8)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2^3,5,23,67|b^2=c,b^a=bc,d^a=d^4,e^a=e^22,f^a=f^66

Conjugacy classes & character table

15413 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 23, 67]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30820 #72296547