D₁₁₁₅₁ #72215842

order	22302
automorphism	Aut(G) order 69849864
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₁₁₁₅₁

Splittings

Fitting, derived, Hall	C₇×C₂₇×C₅₉ ⋊ C₂
Hall	C₂₇×C₅₉ ⋊ D₇
Hall	C₇×C₅₉ ⋊ D₂₇
Hall	C₇×C₂₇ ⋊ D₅₉
Sylow	C₅₉ ⋊ D₁₈₉
Sylow	C₂₇ ⋊ D₄₁₃
Sylow	C₇ ⋊ D₁₅₉₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₇×C₅₉	C₂
Frattini	C₉	D₁₂₃₉
socle	C₃×C₇×C₅₉	D₉
LCS	G ▷ C₇×C₂₇×C₅₉
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 11151

A minimal faithful permutation representation — 2 generators on 93 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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(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)(29,34)(30,33)(31,32)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)

Polycyclic presentation:

2,3,7,59,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^6,d^a=d^58,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

5577 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₁₁₅₁
maximal normal subgroup of	C₂×D₁₁₁₅₁
maximal quotient of	Dic₁₁₁₅₁, C₂×D₁₁₁₅₁

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, 3, 7, 59, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^58);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11151 #72215842