Dic₅₀₄₉ #72216637

order	20196
automorphism	Aut(G) order 29082240
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₀₄₉

Splittings

derived, Hall	C₁₁×C₁₇×C₂₇ ⋊ C₄
Hall	C₁₇×C₂₇ ⋊ Dic₁₁
Hall	C₁₁×C₂₇ ⋊ Dic₁₇
Hall	C₁₁×C₁₇ ⋊ Dic₂₇
Sylow	C₂₇ ⋊ Dic₁₈₇
Sylow	C₁₇ ⋊ Dic₂₉₇
Sylow	C₁₁ ⋊ Dic₄₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₀₄₉
commutator, hypocenter	C₁₁×C₁₇×C₂₇	C₄
Frattini	C₂×C₉	D₅₆₁
Fitting, cosocle	C₂×C₁₁×C₁₇×C₂₇	C₂
socle, prime radical	C₂×C₃×C₁₁×C₁₇	D₉
LCS	G ▷ C₁₁×C₁₇×C₂₇
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 20196

A minimal faithful permutation representation — 2 generators on 59 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(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);(1,3,2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,42)(34,41)(35,40)(36,39)(37,38)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2^2,3,11,17,3^2|a^2=b,c^3=f,f^3=g,c^a=c^2f^2g^2,d^a=d^10,e^a=e^16,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

5052 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₀₉₈, Dic₁₅₁₄₇
maximal normal subgroup of	Dic₁₀₀₉₈
maximal quotient of	Dic₁₅₁₄₇

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, 3, 11, 17, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5049 #72216637