Dic₁₂₂₁₂ #72282715

order	48848
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₂₁₂

Splittings

Hall	C₄₃×C₇₁ ⋊ Q₁₆
Sylow	C₇₁ ⋊ Dic₁₇₂
Sylow	C₄₃ ⋊ Dic₂₈₄

Subgroups

	subgroup	G/H
center	C₂	D₁₂₂₁₂
commutator, cosocle	C₄×C₄₃×C₇₁	C₂²
Frattini	C₄	C₂×D₃₀₅₃
Fitting	C₈×C₄₃×C₇₁	C₂
socle, prime radical	C₂×C₄₃×C₇₁	D₄
hypercenter	C₈	D₃₀₅₃
hypocenter	C₄₃×C₇₁	Q₁₆
LCS	G ▷ C₄×C₄₃×C₇₁ ▷ C₂×C₄₃×C₇₁ ▷ C₄₃×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 130
transitive degree	μ_t(G) = 48848

A minimal faithful permutation representation — 2 generators on 130 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)(60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)

Polycyclic presentation:

2^4,43,71|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^42,f^a=f^70

Conjugacy classes & character table

12215 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, 2, 43, 71]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^42);;
gap> SetConjugate(coll, 6, 1, F.6^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12212 #72282715