Dic₁₂₈₈₂ #72288348

order	51528
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₈₈₂

Splittings

Hall	C₃×C₁₉×C₁₁₃ ⋊ Q₈
Hall	C₁₉×C₁₁₃ ⋊ Dic₆
Hall	C₃×C₁₁₃ ⋊ Dic₃₈
Sylow	C₁₁₃ ⋊ Dic₁₁₄
Hall	C₃×C₁₉ ⋊ Dic₂₂₆
Sylow	C₁₉ ⋊ Dic₆₇₈
Sylow	C₃ ⋊ Dic₄₂₉₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₄₄₁
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₉×C₁₁₃	C₂²
Fitting	C₃×C₄×C₁₉×C₁₁₃	C₂
hypercenter	C₄	D₆₄₄₁
hypocenter	C₃×C₁₉×C₁₁₃	Q₈
LCS	G ▷ C₂×C₃×C₁₉×C₁₁₃ ▷ C₃×C₁₉×C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 143
transitive degree	μ_t(G) = 51528

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

Polycyclic presentation:

2^3,3,19,113|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^18,f^a=f^112

Conjugacy classes & character table

12885 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, 19, 113]);;
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.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^112);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12882 #72288348