Dic₁₄₃₁₀ #72290912

order	57240
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₁₀

Splittings

Hall	C₅×C₂₇×C₅₃ ⋊ Q₈
Hall	C₂₇×C₅₃ ⋊ Dic₁₀
Hall	C₅×C₅₃ ⋊ Dic₅₄
Hall	C₅×C₂₇ ⋊ Dic₁₀₆
Sylow	C₅₃ ⋊ Dic₂₇₀
Sylow	C₂₇ ⋊ Dic₅₃₀
Sylow	C₅ ⋊ Dic₂₈₆₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₁₅₅
commutator, cosocle	C₂×C₅×C₂₇×C₅₃	C₂²
Frattini	C₂×C₉	C₂×D₇₉₅
Fitting	C₄×C₅×C₂₇×C₅₃	C₂
socle, prime radical	C₂×C₃×C₅×C₅₃	C₂×D₉
hypercenter	C₄	D₇₁₅₅
hypocenter	C₅×C₂₇×C₅₃	Q₈
LCS	G ▷ C₂×C₅×C₂₇×C₅₃ ▷ C₅×C₂₇×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 57240

A minimal faithful permutation representation — 2 generators on 93 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35)(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);(1,5,3,7)(2,6,4,8)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(37,40)(38,39)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2^3,3,5,53,3^2|a^2=c,b^2=c,d^3=g,g^3=h,b^a=bc,d^a=d^2g^2h^2,e^a=e^4,f^a=f^52,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

14313 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, 3, 5, 53, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^52);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14310 #72290912