Dic₁₃₃₀₀ #72289017

order	53200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₃₀₀

Splittings

Hall	C₇×C₁₉×C₂₅ ⋊ Q₁₆
Hall	C₁₉×C₂₅ ⋊ Dic₂₈
Hall	C₇×C₂₅ ⋊ Dic₇₆
Hall	C₇×C₁₉ ⋊ Dic₁₀₀
Sylow	C₂₅ ⋊ Dic₅₃₂
Sylow	C₁₉ ⋊ Dic₇₀₀
Sylow	C₇ ⋊ Dic₁₉₀₀

Subgroups

	subgroup	G/H
center	C₂	D₁₃₃₀₀
commutator, cosocle	C₄×C₇×C₁₉×C₂₅	C₂²
Frattini	C₄×C₅	C₂×D₆₆₅
Fitting	C₇×C₈×C₁₉×C₂₅	C₂
socle, prime radical	C₂×C₅×C₇×C₁₉	D₂₀
hypercenter	C₈	D₃₃₂₅
hypocenter	C₇×C₁₉×C₂₅	Q₁₆
LCS	G ▷ C₄×C₇×C₁₉×C₂₅ ▷ C₂×C₇×C₁₉×C₂₅ ▷ C₇×C₁₉×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 53200

A minimal faithful permutation representation — 2 generators on 67 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(43,48)(44,47)(45,46)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Polycyclic presentation:

2^4,5,7,19,5|a^2=d,b^2=cd,c^2=d,e^5=h,b^a=bc,c^a=cd,e^a=e^4h^4,f^a=f^6,g^a=g^18,h^a=h^4

Conjugacy classes & character table

13303 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, 5, 7, 19, 5]);;
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.5^4*F.8^4);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^18);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13300 #72289017