Dic₇₄₅₂ #72236624

order	29808
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₄₅₂

Splittings

Hall	C₂₃×C₈₁ ⋊ Q₁₆
Sylow	C₈₁ ⋊ Dic₉₂
Sylow	C₂₃ ⋊ Dic₃₂₄

Subgroups

	subgroup	G/H
center	C₂	D₇₄₅₂
commutator, cosocle	C₄×C₂₃×C₈₁	C₂²
Frattini	C₄×C₂₇	C₂×D₆₉
Fitting	C₈×C₂₃×C₈₁	C₂
socle, prime radical	C₂×C₃×C₂₃	D₁₀₈
hypercenter	C₈	D₁₈₆₃
hypocenter	C₂₃×C₈₁	Q₁₆
LCS	G ▷ C₄×C₂₃×C₈₁ ▷ C₂×C₂₃×C₈₁ ▷ C₂₃×C₈₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 120
transitive degree	μ_t(G) = 29808

A minimal faithful permutation representation — 2 generators on 120 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97)(98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(99,120)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(109,110)

Polycyclic presentation:

2^4,3,23,3^3|a^2=d,b^2=cd,c^2=d,e^3=g,g^3=h,h^3=i,b^a=bc,c^a=cd,e^a=e^2g^2h^2i^2,f^a=f^22,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^2

Conjugacy classes & character table

7455 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₉₀₄
maximal normal subgroup of	Dic₁₄₉₀₄

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 23, 3, 3, 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.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
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.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7452 #72236624