Dic₄₆₉₂ #72216147

order	18768
automorphism	Aut(G) order 26425344
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₆₉₂

Splittings

Hall	C₃×C₁₇×C₂₃ ⋊ Q₁₆
Hall	C₁₇×C₂₃ ⋊ Dic₁₂
Hall	C₃×C₂₃ ⋊ Dic₆₈
Hall	C₃×C₁₇ ⋊ Dic₉₂
Sylow	C₂₃ ⋊ Dic₂₀₄
Sylow	C₁₇ ⋊ Dic₂₇₆
Sylow	C₃ ⋊ Dic₁₅₆₄

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, prime radical	C₂×C₃×C₁₇×C₂₃	D₄
hypercenter	C₈	D₁₁₇₃
hypocenter	C₃×C₁₇×C₂₃	Q₁₆
LCS	G ▷ C₃×C₄×C₁₇×C₂₃ ▷ C₂×C₃×C₁₇×C₂₃ ▷ C₃×C₁₇×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 18768

A minimal faithful permutation representation — 2 generators on 59 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2^4,3,17,23|a^2=d,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	Dic₉₃₈₄, Dic₁₄₀₇₆
maximal normal subgroup of	Dic₉₃₈₄
maximal quotient of	Dic₁₄₀₇₆

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, 1, F.4);;
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);

Dic4692 #72216147