Dic₉₉₀₆ #72274028

order	39624
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₉₉₀₆

Splittings

Hall	C₃×C₁₃×C₁₂₇ ⋊ Q₈
Hall	C₁₃×C₁₂₇ ⋊ Dic₆
Hall	C₃×C₁₂₇ ⋊ Dic₂₆
Sylow	C₁₂₇ ⋊ Dic₇₈
Hall	C₃×C₁₃ ⋊ Dic₂₅₄
Sylow	C₁₃ ⋊ Dic₇₆₂
Sylow	C₃ ⋊ Dic₃₃₀₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₄₉₅₃
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₃×C₁₂₇	C₂²
Fitting	C₃×C₄×C₁₃×C₁₂₇	C₂
hypercenter	C₄	D₄₉₅₃
hypocenter	C₃×C₁₃×C₁₂₇	Q₈
LCS	G ▷ C₂×C₃×C₁₃×C₁₂₇ ▷ C₃×C₁₃×C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 151
transitive degree	μ_t(G) = 39624

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

Polycyclic presentation:

2^3,3,13,127|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^12,f^a=f^126

Conjugacy classes & character table

9909 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, 3, 13, 127]);;
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.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^126);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic9906 #72274028