D₃₀₈₅₆ #72296885

order	61712
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₈₅₆

Splittings

Fitting	C₇×C₈×C₁₉×C₂₉ ⋊ C₂
Hall	C₇×C₁₉×C₂₉ ⋊ D₈
Hall	C₁₉×C₂₉ ⋊ D₅₆
Hall	C₇×C₂₉ ⋊ D₁₅₂
Hall	C₇×C₁₉ ⋊ D₂₃₂
Sylow	C₂₉ ⋊ D₁₀₆₄
Sylow	C₁₉ ⋊ D₁₆₂₄
Sylow	C₇ ⋊ D₄₄₀₈

Subgroups

	subgroup	G/H
center	C₂	D₁₅₄₂₈
commutator, cosocle	C₄×C₇×C₁₉×C₂₉	C₂²
Frattini	C₄	C₂×D₃₈₅₇
Fitting	C₇×C₈×C₁₉×C₂₉	C₂
socle	C₂×C₇×C₁₉×C₂₉	D₄
hypercenter	C₈	D₃₈₅₇
hypocenter	C₇×C₁₉×C₂₉	D₈
LCS	G ▷ C₄×C₇×C₁₉×C₂₉ ▷ C₂×C₇×C₁₉×C₂₉ ▷ C₇×C₁₉×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 30856

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);(2,8)(3,7)(4,6)(10,15)(11,14)(12,13)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2^4,7,19,29|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^18,g^a=g^28

Conjugacy classes & character table

15431 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 7, 19, 29]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30856 #72296885