D₂₄₁₄₀ #72286282

order	48280
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₄₁₄₀

Splittings

Fitting	C₄×C₅×C₁₇×C₇₁ ⋊ C₂
Hall	C₅×C₁₇×C₇₁ ⋊ D₄
Hall	C₁₇×C₇₁ ⋊ D₂₀
Hall	C₅×C₇₁ ⋊ D₆₈
Hall	C₅×C₁₇ ⋊ D₂₈₄
Sylow	C₇₁ ⋊ D₃₄₀
Sylow	C₁₇ ⋊ D₁₄₂₀
Sylow	C₅ ⋊ D₄₈₂₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₀₃₅
commutator, socle, cosocle	C₂×C₅×C₁₇×C₇₁	C₂²
Fitting	C₄×C₅×C₁₇×C₇₁	C₂
hypercenter	C₄	D₆₀₃₅
hypocenter	C₅×C₁₇×C₇₁	D₄
LCS	G ▷ C₂×C₅×C₁₇×C₇₁ ▷ C₅×C₁₇×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 24140

A minimal faithful permutation representation — 2 generators on 97 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);(2,4)(6,9)(7,8)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)

Polycyclic presentation:

2^3,5,17,71|b^2=c,b^a=bc,d^a=d^4,e^a=e^16,f^a=f^70

Conjugacy classes & character table

12073 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, 17, 71]);;
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.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D24140 #72286282