Dic₁₄₈₄₀ #72292131

order	59360
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₈₄₀

Splittings

Hall	C₅×C₇×C₅₃ ⋊ Q₃₂
Hall	C₇×C₅₃ ⋊ Dic₄₀
Hall	C₅×C₅₃ ⋊ Dic₅₆
Sylow	C₅₃ ⋊ Dic₂₈₀
Hall	C₅×C₇ ⋊ Dic₄₂₄
Sylow	C₇ ⋊ Dic₂₁₂₀
Sylow	C₅ ⋊ Dic₂₉₆₈

Subgroups

	subgroup	G/H
center	C₂	D₁₄₈₄₀
commutator, cosocle	C₅×C₇×C₈×C₅₃	C₂²
Frattini	C₈	C₂×D₁₈₅₅
Fitting	C₅×C₇×C₁₆×C₅₃	C₂
socle, prime radical	C₂×C₅×C₇×C₅₃	D₈
hypercenter	C₁₆	D₁₈₅₅
hypocenter	C₅×C₇×C₅₃	Q₃₂
LCS	G ▷ C₅×C₇×C₈×C₅₃ ▷ C₄×C₅×C₇×C₅₃ ▷ C₂×C₅×C₇×C₅₃ ▷ C₅×C₇×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 59360

A minimal faithful permutation representation — 2 generators on 97 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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(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)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2^5,5,7,53|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^52

Conjugacy classes & character table

14843 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, 2, 5, 7, 53]);;
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.8^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14840 #72292131