Dic₆₄₄₀ #72236096

order	25760
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₄₄₀

Splittings

Hall	C₅×C₇×C₂₃ ⋊ Q₃₂
Hall	C₇×C₂₃ ⋊ Dic₄₀
Hall	C₅×C₂₃ ⋊ Dic₅₆
Hall	C₅×C₇ ⋊ Dic₁₈₄
Sylow	C₂₃ ⋊ Dic₂₈₀
Sylow	C₇ ⋊ Dic₉₂₀
Sylow	C₅ ⋊ Dic₁₂₈₈

Subgroups

	subgroup	G/H
center	C₂	D₆₄₄₀
commutator, cosocle	C₅×C₇×C₈×C₂₃	C₂²
Frattini	C₈	C₂×D₈₀₅
Fitting	C₅×C₇×C₁₆×C₂₃	C₂
socle, prime radical	C₂×C₅×C₇×C₂₃	D₈
hypercenter	C₁₆	D₈₀₅
hypocenter	C₅×C₇×C₂₃	Q₃₂
LCS	G ▷ C₅×C₇×C₈×C₂₃ ▷ C₄×C₅×C₇×C₂₃ ▷ C₂×C₅×C₇×C₂₃ ▷ C₅×C₇×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 25760

A minimal faithful permutation representation — 2 generators on 67 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,37)(35,36)(39,44)(40,43)(41,42)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2^5,5,7,23|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^4,g^a=g^6,h^a=h^22

Conjugacy classes & character table

6443 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₈₈₀
maximal normal subgroup of	Dic₁₂₈₈₀

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 7, 23]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6440 #72236096