Dic₁₈₁₂ #72212232

order	7248
automorphism	Aut(G) order 4348800
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₈₁₂

Splittings

Hall	C₃×C₁₅₁ ⋊ Q₁₆
Sylow	C₁₅₁ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₆₀₄

Subgroups

	subgroup	G/H
center	C₂	D₁₈₁₂
commutator, cosocle	C₃×C₄×C₁₅₁	C₂²
Frattini	C₄	C₂×D₄₅₃
Fitting	C₃×C₈×C₁₅₁	C₂
socle, prime radical	C₂×C₃×C₁₅₁	D₄
hypercenter	C₈	D₄₅₃
hypocenter	C₃×C₁₅₁	Q₁₆
LCS	G ▷ C₃×C₄×C₁₅₁ ▷ C₂×C₃×C₁₅₁ ▷ C₃×C₁₅₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 170
transitive degree	μ_t(G) = 7248

A minimal faithful permutation representation — 2 generators on 170 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,170)(22,169)(23,168)(24,167)(25,166)(26,165)(27,164)(28,163)(29,162)(30,161)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,150)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(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:

2^4,3,151|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^150

Conjugacy classes & character table

1815 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃₆₂₄, Dic₅₄₃₆, Dic₉₀₆₀, …
maximal normal subgroup of	Dic₃₆₂₄
maximal quotient of	Dic₅₄₃₆, Dic₉₀₆₀, Dic₁₂₆₈₄

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, 3, 151]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^150);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1812 #72212232