D₁₄₅₃₆ #72235828

order	29072
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₅₃₆

Splittings

Fitting	C₈×C₂₃×C₇₉ ⋊ C₂
Hall	C₂₃×C₇₉ ⋊ D₈
Sylow	C₇₉ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₆₃₂

Subgroups

	subgroup	G/H
center	C₂	D₇₂₆₈
commutator, cosocle	C₄×C₂₃×C₇₉	C₂²
Frattini	C₄	C₂×D₁₈₁₇
Fitting	C₈×C₂₃×C₇₉	C₂
socle	C₂×C₂₃×C₇₉	D₄
hypercenter	C₈	D₁₈₁₇
hypocenter	C₂₃×C₇₉	D₈
LCS	G ▷ C₄×C₂₃×C₇₉ ▷ C₂×C₂₃×C₇₉ ▷ C₂₃×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 110
transitive degree	μ_t(G) = 14536

A minimal faithful permutation representation — 2 generators on 110 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)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110);(2,8)(3,7)(4,6)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2^4,23,79|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^22,f^a=f^78

Conjugacy classes & character table

7271 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₉₀₇₂
maximal normal subgroup of	D₂₉₀₇₂
maximal quotient of	Dic₁₄₅₃₆, D₂₉₀₇₂

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, 23, 79]);;
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.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^78);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14536 #72235828