D₃₁₂₄₈ #72300626

order	62496
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₂₄₈

Splittings

Fitting	C₇×C₉×C₁₆×C₃₁ ⋊ C₂
Hall	C₇×C₉×C₃₁ ⋊ D₁₆
Hall	C₉×C₃₁ ⋊ D₁₁₂
Hall	C₇×C₃₁ ⋊ D₁₄₄
Hall	C₇×C₉ ⋊ D₄₉₆
Sylow	C₃₁ ⋊ D₁₀₀₈
Sylow	C₉ ⋊ D₃₄₇₂
Sylow	C₇ ⋊ D₄₄₆₄

Subgroups

	subgroup	G/H
center	C₂	D₁₅₆₂₄
commutator, cosocle	C₇×C₈×C₉×C₃₁	C₂²
Frattini	C₃×C₈	C₂×D₆₅₁
Fitting	C₇×C₉×C₁₆×C₃₁	C₂
socle	C₂×C₃×C₇×C₃₁	D₂₄
hypercenter	C₁₆	D₁₉₅₃
hypocenter	C₇×C₉×C₃₁	D₁₆
LCS	G ▷ C₇×C₈×C₉×C₃₁ ▷ C₄×C₇×C₉×C₃₁ ▷ C₂×C₇×C₉×C₃₁ ▷ C₇×C₉×C₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

168 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 31248

A minimal faithful permutation representation — 2 generators on 63 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,25)(19,24)(20,23)(21,22)(27,32)(28,31)(29,30)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2^5,3,7,31,3|b^2=cd,c^2=de,d^2=e,f^3=i,b^a=bc,c^a=cd,d^a=de,f^a=f^2i^2,g^a=g^6,h^a=h^30,i^a=i^2

Conjugacy classes & character table

15627 conjugacy classes. The character table is too large to display.

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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 7, 31, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^30);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);