Dic₃₀₅₈ #72213305

order	12232
automorphism	Aut(G) order 16880160
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₀₅₈

Splittings

Hall	C₁₁×C₁₃₉ ⋊ Q₈
Sylow	C₁₃₉ ⋊ Dic₂₂
Sylow	C₁₁ ⋊ Dic₂₇₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₅₂₉
commutator, socle, prime radical, cosocle	C₂×C₁₁×C₁₃₉	C₂²
Fitting	C₄×C₁₁×C₁₃₉	C₂
hypercenter	C₄	D₁₅₂₉
hypocenter	C₁₁×C₁₃₉	Q₈
LCS	G ▷ C₂×C₁₁×C₁₃₉ ▷ C₁₁×C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 158
transitive degree	μ_t(G) = 12232

A minimal faithful permutation representation — 2 generators on 158 points:

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

Polycyclic presentation:

2^3,11,139|a^2=c,b^2=c,b^a=bc,d^a=d^10,e^a=e^138

Conjugacy classes & character table

3061 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₁₁₆, Dic₉₁₇₄, Dic₁₅₂₉₀
maximal normal subgroup of	Dic₆₁₁₆
maximal quotient of	Dic₉₁₇₄, Dic₁₅₂₉₀

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 11, 139]);;
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.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^138);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3058 #72213305