Dic₂₄₆₆ #72212751

order	9864
automorphism	Aut(G) order 8049024
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₄₆₆

Splittings

Hall	C₉×C₁₃₇ ⋊ Q₈
Sylow	C₁₃₇ ⋊ Dic₁₈
Sylow	C₉ ⋊ Dic₂₇₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₂₃₃
commutator, cosocle	C₂×C₉×C₁₃₇	C₂²
Frattini	C₂×C₃	C₂×D₄₁₁
Fitting	C₄×C₉×C₁₃₇	C₂
socle, prime radical	C₂×C₃×C₁₃₇	C₂×D₃
hypercenter	C₄	D₁₂₃₃
hypocenter	C₉×C₁₃₇	Q₈
LCS	G ▷ C₂×C₉×C₁₃₇ ▷ C₉×C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 154
transitive degree	μ_t(G) = 9864

A minimal faithful permutation representation — 2 generators on 154 points:

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

Polycyclic presentation:

2^3,3,137,3|a^2=c,b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^136,f^a=f^2

Conjugacy classes & character table

2469 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₄₉₃₂, Dic₇₃₉₈, Dic₁₂₃₃₀
maximal normal subgroup of	Dic₄₉₃₂
maximal quotient of	Dic₇₃₉₈, Dic₁₂₃₃₀

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, 3, 137, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^136);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2466 #72212751