D₁₁₄₈₄ #72223909

order	22968
automorphism	Aut(G) order 38586240
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₄₈₄

Splittings

Fitting	C₄×C₉×C₁₁×C₂₉ ⋊ C₂
Hall	C₉×C₁₁×C₂₉ ⋊ D₄
Hall	C₁₁×C₂₉ ⋊ D₃₆
Hall	C₉×C₂₉ ⋊ D₄₄
Hall	C₉×C₁₁ ⋊ D₁₁₆
Sylow	C₂₉ ⋊ D₃₉₆
Sylow	C₁₁ ⋊ D₁₀₄₄
Sylow	C₉ ⋊ D₁₂₇₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₈₇₁
commutator, cosocle	C₂×C₉×C₁₁×C₂₉	C₂²
Frattini	C₂×C₃	C₂×D₉₅₇
Fitting	C₄×C₉×C₁₁×C₂₉	C₂
socle	C₂×C₃×C₁₁×C₂₉	C₂×D₃
hypercenter	C₄	D₂₈₇₁
hypocenter	C₉×C₁₁×C₂₉	D₄
LCS	G ▷ C₂×C₉×C₁₁×C₂₉ ▷ C₉×C₁₁×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 11484

A minimal faithful permutation representation — 2 generators on 53 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,24)(16,23)(17,22)(18,21)(19,20)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)

Polycyclic presentation:

2^3,3,11,29,3|b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^10,f^a=f^28,g^a=g^2

Conjugacy classes & character table

5745 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₉₆₈
maximal normal subgroup of	D₂₂₉₆₈
maximal quotient of	Dic₁₁₄₈₄, D₂₂₉₆₈

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 11, 29, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11484 #72223909