D₂₆₇₇₅ #72290119

order	53550
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₇₇₅

Splittings

Fitting, derived, Hall	C₇×C₉×C₁₇×C₂₅ ⋊ C₂
Hall	C₉×C₁₇×C₂₅ ⋊ D₇
Hall	C₇×C₁₇×C₂₅ ⋊ D₉
Hall	C₇×C₉×C₂₅ ⋊ D₁₇
Hall	C₇×C₉×C₁₇ ⋊ D₂₅
Hall	C₁₇×C₂₅ ⋊ D₆₃
Hall	C₉×C₂₅ ⋊ D₁₁₉
Hall	C₇×C₂₅ ⋊ D₁₅₃
Hall	C₉×C₁₇ ⋊ D₁₇₅
Hall	C₇×C₁₇ ⋊ D₂₂₅
Hall	C₇×C₉ ⋊ D₄₂₅
Sylow	C₂₅ ⋊ D₁₀₇₁
Sylow	C₁₇ ⋊ D₁₅₇₅
Sylow	C₉ ⋊ D₂₉₇₅
Sylow	C₇ ⋊ D₃₈₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₉×C₁₇×C₂₅	C₂
Frattini	C₃×C₅	D₁₇₈₅
socle	C₃×C₅×C₇×C₁₇	D₁₅
LCS	G ▷ C₇×C₉×C₁₇×C₂₅
UCS	C₁

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 58
transitive degree	μ_t(G) = 26775

A minimal faithful permutation representation — 2 generators on 58 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);(2,9)(3,8)(4,7)(5,6)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(36,41)(37,40)(38,39)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2,3,5,7,17,3,5|b^3=f,c^5=g,b^a=b^2f^2,c^a=c^4g^4,d^a=d^6,e^a=e^16,f^a=f^2,g^a=g^4

Conjugacy classes & character table

13389 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, 3, 5, 7, 17, 3, 5]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^4*F.7^4);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26775 #72290119