D₁₄₁₅₂ #72235682

order	28304
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₁₅₂

Splittings

Fitting	C₈×C₂₉×C₆₁ ⋊ C₂
Hall	C₂₉×C₆₁ ⋊ D₈
Sylow	C₆₁ ⋊ D₂₃₂
Sylow	C₂₉ ⋊ D₄₈₈

Subgroups

	subgroup	G/H
center	C₂	D₇₀₇₆
commutator, cosocle	C₄×C₂₉×C₆₁	C₂²
Frattini	C₄	C₂×D₁₇₆₉
Fitting	C₈×C₂₉×C₆₁	C₂
socle	C₂×C₂₉×C₆₁	D₄
hypercenter	C₈	D₁₇₆₉
hypocenter	C₂₉×C₆₁	D₈
LCS	G ▷ C₄×C₂₉×C₆₁ ▷ C₂×C₂₉×C₆₁ ▷ C₂₉×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 98
transitive degree	μ_t(G) = 14152

A minimal faithful permutation representation — 2 generators on 98 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);(2,8)(3,7)(4,6)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2^4,29,61|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^28,f^a=f^60

Conjugacy classes & character table

7079 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₈₃₀₄
maximal normal subgroup of	D₂₈₃₀₄
maximal quotient of	Dic₁₄₁₅₂, D₂₈₃₀₄

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, 2, 29, 61]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^28);;
gap> SetConjugate(coll, 6, 1, F.6^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14152 #72235682