Dic₁₆₃₃₀ #72301580

order	65320
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₃₃₀

Splittings

Hall	C₅×C₂₃×C₇₁ ⋊ Q₈
Hall	C₂₃×C₇₁ ⋊ Dic₁₀
Hall	C₅×C₇₁ ⋊ Dic₄₆
Hall	C₅×C₂₃ ⋊ Dic₁₄₂
Sylow	C₇₁ ⋊ Dic₂₃₀
Sylow	C₂₃ ⋊ Dic₇₁₀
Sylow	C₅ ⋊ Dic₃₂₆₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₈₁₆₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₂₃×C₇₁	C₂²
Fitting	C₄×C₅×C₂₃×C₇₁	C₂
hypercenter	C₄	D₈₁₆₅
hypocenter	C₅×C₂₃×C₇₁	Q₈
LCS	G ▷ C₂×C₅×C₂₃×C₇₁ ▷ C₅×C₂₃×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 65320

A minimal faithful permutation representation — 2 generators on 107 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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

2^3,5,23,71|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^22,f^a=f^70

Conjugacy classes & character table

16333 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 23, 71]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic16330 #72301580