D₁₀₅₅₆ #72215440

order	21112
automorphism	Aut(G) order 42561792
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₅₅₆

Splittings

Fitting	C₄×C₇×C₁₃×C₂₉ ⋊ C₂
Hall	C₇×C₁₃×C₂₉ ⋊ D₄
Hall	C₁₃×C₂₉ ⋊ D₂₈
Hall	C₇×C₂₉ ⋊ D₅₂
Hall	C₇×C₁₃ ⋊ D₁₁₆
Sylow	C₂₉ ⋊ D₃₆₄
Sylow	C₁₃ ⋊ D₈₁₂
Sylow	C₇ ⋊ D₁₅₀₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₆₃₉
commutator, socle, cosocle	C₂×C₇×C₁₃×C₂₉	C₂²
Fitting	C₄×C₇×C₁₃×C₂₉	C₂
hypercenter	C₄	D₂₆₃₉
hypocenter	C₇×C₁₃×C₂₉	D₄
LCS	G ▷ C₂×C₇×C₁₃×C₂₉ ▷ C₇×C₁₃×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 10556

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)(6,11)(7,10)(8,9)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(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:

2^3,7,13,29|b^2=c,b^a=bc,d^a=d^6,e^a=e^12,f^a=f^28

Conjugacy classes & character table

5281 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₁₁₁₂, D₃₁₆₆₈
maximal normal subgroup of	D₂₁₁₁₂
maximal quotient of	Dic₁₀₅₅₆, D₂₁₁₁₂, D₃₁₆₆₈

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, 7, 13, 29]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D10556 #72215440