D₁₂₂₈₅ #72224075

order	24570
automorphism	Aut(G) order 63685440
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₁₂₂₈₅

Splittings

Fitting, derived, Hall	C₅×C₇×C₁₃×C₂₇ ⋊ C₂
Hall	C₇×C₁₃×C₂₇ ⋊ D₅
Hall	C₅×C₁₃×C₂₇ ⋊ D₇
Hall	C₅×C₇×C₂₇ ⋊ D₁₃
Hall	C₅×C₇×C₁₃ ⋊ D₂₇
Hall	C₁₃×C₂₇ ⋊ D₃₅
Hall	C₇×C₂₇ ⋊ D₆₅
Hall	C₅×C₂₇ ⋊ D₉₁
Hall	C₇×C₁₃ ⋊ D₁₃₅
Hall	C₅×C₁₃ ⋊ D₁₈₉
Hall	C₅×C₇ ⋊ D₃₅₁
Sylow	C₂₇ ⋊ D₄₅₅
Sylow	C₁₃ ⋊ D₉₄₅
Sylow	C₇ ⋊ D₁₇₅₅
Sylow	C₅ ⋊ D₂₄₅₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₇×C₁₃×C₂₇	C₂
Frattini	C₉	D₁₃₆₅
socle	C₃×C₅×C₇×C₁₃	D₉
LCS	G ▷ C₅×C₇×C₁₃×C₂₇
UCS	C₁

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 12285

A minimal faithful permutation representation — 2 generators on 52 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);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,32)(30,31)(34,39)(35,38)(36,37)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2,3,5,7,13,3^2|b^3=f,f^3=g,b^a=b^2f^2g^2,c^a=c^4,d^a=d^6,e^a=e^12,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

6144 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₂₂₈₅
maximal normal subgroup of	C₂×D₁₂₂₈₅
maximal quotient of	Dic₁₂₂₈₅, C₂×D₁₂₂₈₅

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

D12285 #72224075