Dic₅₂₄₄ #72216997

order	20976
automorphism	Aut(G) order 33225984
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₂₄₄

Splittings

Hall	C₃×C₁₉×C₂₃ ⋊ Q₁₆
Hall	C₁₉×C₂₃ ⋊ Dic₁₂
Hall	C₃×C₂₃ ⋊ Dic₇₆
Hall	C₃×C₁₉ ⋊ Dic₉₂
Sylow	C₂₃ ⋊ Dic₂₂₈
Sylow	C₁₉ ⋊ Dic₂₇₆
Sylow	C₃ ⋊ Dic₁₇₄₈

Subgroups

	subgroup	G/H
center	C₂	D₅₂₄₄
commutator, cosocle	C₃×C₄×C₁₉×C₂₃	C₂²
Frattini	C₄	C₂×D₁₃₁₁
Fitting	C₃×C₈×C₁₉×C₂₃	C₂
socle, prime radical	C₂×C₃×C₁₉×C₂₃	D₄
hypercenter	C₈	D₁₃₁₁
hypocenter	C₃×C₁₉×C₂₃	Q₁₆
LCS	G ▷ C₃×C₄×C₁₉×C₂₃ ▷ C₂×C₃×C₁₉×C₂₃ ▷ C₃×C₁₉×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 20976

A minimal faithful permutation representation — 2 generators on 61 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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2^4,3,19,23|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^18,g^a=g^22

Conjugacy classes & character table

5247 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₄₈₈, Dic₁₅₇₃₂
maximal normal subgroup of	Dic₁₀₄₈₈
maximal quotient of	Dic₁₅₇₃₂

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, 19, 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.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5244 #72216997