D₉₃₈₄ #72214307

order	18768
automorphism	Aut(G) order 26425344
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₃₈₄

Splittings

Fitting	C₃×C₈×C₁₇×C₂₃ ⋊ C₂
Hall	C₃×C₁₇×C₂₃ ⋊ D₈
Hall	C₁₇×C₂₃ ⋊ D₂₄
Hall	C₃×C₂₃ ⋊ D₁₃₆
Hall	C₃×C₁₇ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₄₀₈
Sylow	C₁₇ ⋊ D₅₅₂
Sylow	C₃ ⋊ D₃₁₂₈

Subgroups

	subgroup	G/H
center	C₂	D₄₆₉₂
commutator, cosocle	C₃×C₄×C₁₇×C₂₃	C₂²
Frattini	C₄	C₂×D₁₁₇₃
Fitting	C₃×C₈×C₁₇×C₂₃	C₂
socle	C₂×C₃×C₁₇×C₂₃	D₄
hypercenter	C₈	D₁₁₇₃
hypocenter	C₃×C₁₇×C₂₃	D₈
LCS	G ▷ C₃×C₄×C₁₇×C₂₃ ▷ C₂×C₃×C₁₇×C₂₃ ▷ C₃×C₁₇×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 9384

A minimal faithful permutation representation — 2 generators on 51 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);(2,8)(3,7)(4,6)(10,11)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Polycyclic presentation:

2^4,3,17,23|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^16,g^a=g^22

Conjugacy classes & character table

4695 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₈₇₆₈, D₂₈₁₅₂
maximal normal subgroup of	D₁₈₇₆₈
maximal quotient of	Dic₉₃₈₄, D₁₈₇₆₈, D₂₈₁₅₂

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, 3, 17, 23]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9384 #72214307