D₈₇₄₀ #72213615

order	17480
automorphism	Aut(G) order 27688320
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₇₄₀

Splittings

Fitting	C₄×C₅×C₁₉×C₂₃ ⋊ C₂
Hall	C₅×C₁₉×C₂₃ ⋊ D₄
Hall	C₁₉×C₂₃ ⋊ D₂₀
Hall	C₅×C₂₃ ⋊ D₇₆
Hall	C₅×C₁₉ ⋊ D₉₂
Sylow	C₂₃ ⋊ D₃₈₀
Sylow	C₁₉ ⋊ D₄₆₀
Sylow	C₅ ⋊ D₁₇₄₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁₈₅
commutator, socle, cosocle	C₂×C₅×C₁₉×C₂₃	C₂²
Fitting	C₄×C₅×C₁₉×C₂₃	C₂
hypercenter	C₄	D₂₁₈₅
hypocenter	C₅×C₁₉×C₂₃	D₄
LCS	G ▷ C₂×C₅×C₁₉×C₂₃ ▷ C₅×C₁₉×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 8740

A minimal faithful permutation representation — 2 generators on 51 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);(2,4)(6,9)(7,8)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Polycyclic presentation:

2^3,5,19,23|b^2=c,b^a=bc,d^a=d^4,e^a=e^18,f^a=f^22

Conjugacy classes & character table

4373 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₄₈₀, D₂₆₂₂₀
maximal normal subgroup of	D₁₇₄₈₀
maximal quotient of	Dic₈₇₄₀, D₁₇₄₈₀, D₂₆₂₂₀

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, 19, 23]);;
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.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8740 #72213615