D₁₁₉₇₂ #72224003

order	23944
automorphism	Aut(G) order 68958720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₉₇₂

Splittings

Fitting	C₄×C₄₁×C₇₃ ⋊ C₂
Hall	C₄₁×C₇₃ ⋊ D₄
Sylow	C₇₃ ⋊ D₁₆₄
Sylow	C₄₁ ⋊ D₂₉₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₉₉₃
commutator, socle, cosocle	C₂×C₄₁×C₇₃	C₂²
Fitting	C₄×C₄₁×C₇₃	C₂
hypercenter	C₄	D₂₉₉₃
hypocenter	C₄₁×C₇₃	D₄
LCS	G ▷ C₂×C₄₁×C₇₃ ▷ C₄₁×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 118
transitive degree	μ_t(G) = 11972

A minimal faithful permutation representation — 2 generators on 118 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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118);(2,4)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)

Polycyclic presentation:

2^3,41,73|b^2=c,b^a=bc,d^a=d^40,e^a=e^72

Conjugacy classes & character table

5989 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₉₄₄
maximal normal subgroup of	D₂₃₉₄₄
maximal quotient of	Dic₁₁₉₇₂, D₂₃₉₄₄

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 41, 73]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^40);;
gap> SetConjugate(coll, 5, 1, F.5^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11972 #72224003