D₉₅₈₈ #72214745

order	19176
automorphism	Aut(G) order 28227072
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₅₈₈

Splittings

Fitting	C₃×C₄×C₁₇×C₄₇ ⋊ C₂
Hall	C₃×C₁₇×C₄₇ ⋊ D₄
Hall	C₁₇×C₄₇ ⋊ D₁₂
Hall	C₃×C₄₇ ⋊ D₆₈
Hall	C₃×C₁₇ ⋊ D₁₈₈
Sylow	C₄₇ ⋊ D₂₀₄
Sylow	C₁₇ ⋊ D₅₆₄
Sylow	C₃ ⋊ D₃₁₉₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₃₉₇
commutator, socle, cosocle	C₂×C₃×C₁₇×C₄₇	C₂²
Fitting	C₃×C₄×C₁₇×C₄₇	C₂
hypercenter	C₄	D₂₃₉₇
hypocenter	C₃×C₁₇×C₄₇	D₄
LCS	G ▷ C₂×C₃×C₁₇×C₄₇ ▷ C₃×C₁₇×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 9588

A minimal faithful permutation representation — 2 generators on 71 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);(2,4)(6,7)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(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:

2^3,3,17,47|b^2=c,b^a=bc,d^a=d^2,e^a=e^16,f^a=f^46

Conjugacy classes & character table

4797 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₉₁₇₆, D₂₈₇₆₄
maximal normal subgroup of	D₁₉₁₇₆
maximal quotient of	Dic₉₅₈₈, D₁₉₁₇₆, D₂₈₇₆₄

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, 2, 2, 3, 17, 47]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9588 #72214745