D₆₉₉₆ #72212497

order	13992
automorphism	Aut(G) order 14551680
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₆₉₉₆

Splittings

Fitting	C₃×C₄×C₁₁×C₅₃ ⋊ C₂
Hall	C₃×C₁₁×C₅₃ ⋊ D₄
Hall	C₁₁×C₅₃ ⋊ D₁₂
Hall	C₃×C₅₃ ⋊ D₄₄
Sylow	C₅₃ ⋊ D₁₃₂
Hall	C₃×C₁₁ ⋊ D₂₁₂
Sylow	C₁₁ ⋊ D₆₃₆
Sylow	C₃ ⋊ D₂₃₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₇₄₉
commutator, socle, cosocle	C₂×C₃×C₁₁×C₅₃	C₂²
Fitting	C₃×C₄×C₁₁×C₅₃	C₂
hypercenter	C₄	D₁₇₄₉
hypocenter	C₃×C₁₁×C₅₃	D₄
LCS	G ▷ C₂×C₃×C₁₁×C₅₃ ▷ C₃×C₁₁×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 6996

A minimal faithful permutation representation — 2 generators on 71 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);(2,4)(6,7)(9,18)(10,17)(11,16)(12,15)(13,14)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^3,3,11,53|b^2=c,b^a=bc,d^a=d^2,e^a=e^10,f^a=f^52

Conjugacy classes & character table

3501 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₃₉₉₂, D₂₀₉₈₈
maximal normal subgroup of	D₁₃₉₉₂
maximal quotient of	D₁₃₉₉₂, Dic₆₉₉₆, D₂₀₉₈₈

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, 3, 11, 53]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D6996 #72212497