Dic₁₅₃₇₂ #72293422

order	61488
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₃₇₂

Splittings

Hall	C₇×C₉×C₆₁ ⋊ Q₁₆
Hall	C₉×C₆₁ ⋊ Dic₂₈
Hall	C₇×C₆₁ ⋊ Dic₃₆
Hall	C₇×C₉ ⋊ Dic₂₄₄
Sylow	C₆₁ ⋊ Dic₂₅₂
Sylow	C₉ ⋊ Dic₁₇₀₈
Sylow	C₇ ⋊ Dic₂₁₉₆

Subgroups

	subgroup	G/H
center	C₂	D₁₅₃₇₂
commutator, cosocle	C₄×C₇×C₉×C₆₁	C₂²
Frattini	C₃×C₄	C₂×D₁₂₈₁
Fitting	C₇×C₈×C₉×C₆₁	C₂
socle, prime radical	C₂×C₃×C₇×C₆₁	D₁₂
hypercenter	C₈	D₃₈₄₃
hypocenter	C₇×C₉×C₆₁	Q₁₆
LCS	G ▷ C₄×C₇×C₉×C₆₁ ▷ C₂×C₇×C₉×C₆₁ ▷ C₇×C₉×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 61488

A minimal faithful permutation representation — 2 generators on 93 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)(27,32)(28,31)(29,30)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2^4,3,7,61,3|a^2=d,b^2=cd,c^2=d,e^3=h,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^6,g^a=g^60,h^a=h^2

Conjugacy classes & character table

15375 conjugacy classes. The character table is too large to display.

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, 3, 7, 61, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
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*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^60);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15372 #72293422