Dic₃₈₂₂ #72215074

order	15288
automorphism	Aut(G) order 15410304
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₈₂₂

Splittings

Hall	C₃×C₁₃×C₄₉ ⋊ Q₈
Hall	C₁₃×C₄₉ ⋊ Dic₆
Hall	C₃×C₄₉ ⋊ Dic₂₆
Sylow	C₄₉ ⋊ Dic₇₈
Hall	C₃×C₁₃ ⋊ Dic₉₈
Sylow	C₁₃ ⋊ Dic₂₉₄
Sylow	C₃ ⋊ Dic₁₂₇₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₉₁₁
commutator, cosocle	C₂×C₃×C₁₃×C₄₉	C₂²
Frattini	C₂×C₇	C₂×D₂₇₃
Fitting	C₃×C₄×C₁₃×C₄₉	C₂
socle, prime radical	C₂×C₃×C₇×C₁₃	C₂×D₇
hypercenter	C₄	D₁₉₁₁
hypocenter	C₃×C₁₃×C₄₉	Q₈
LCS	G ▷ C₂×C₃×C₁₃×C₄₉ ▷ C₃×C₁₃×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 15288

A minimal faithful permutation representation — 2 generators on 73 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);(1,5,3,7)(2,6,4,8)(10,11)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2^3,3,7,13,7|a^2=c,b^2=c,e^7=g,b^a=bc,d^a=d^2,e^a=e^6g^6,f^a=f^12,g^a=g^6

Conjugacy classes & character table

3825 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₆₄₄, Dic₁₁₄₆₆
maximal normal subgroup of	Dic₇₆₄₄
maximal quotient of	Dic₁₁₄₆₆

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 7, 13, 7]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6*F.7^6);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3822 #72215074