Dic₂₅₇₄ #72212804

order	10296
automorphism	Aut(G) order 7413120
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₅₇₄

Splittings

Hall	C₉×C₁₁×C₁₃ ⋊ Q₈
Hall	C₁₁×C₁₃ ⋊ Dic₁₈
Hall	C₉×C₁₃ ⋊ Dic₂₂
Hall	C₉×C₁₁ ⋊ Dic₂₆
Sylow	C₁₃ ⋊ Dic₁₉₈
Sylow	C₁₁ ⋊ Dic₂₃₄
Sylow	C₉ ⋊ Dic₂₈₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₂₈₇
commutator, cosocle	C₂×C₉×C₁₁×C₁₃	C₂²
Frattini	C₂×C₃	C₂×D₄₂₉
Fitting	C₄×C₉×C₁₁×C₁₃	C₂
socle, prime radical	C₂×C₃×C₁₁×C₁₃	C₂×D₃
hypercenter	C₄	D₁₂₈₇
hypocenter	C₉×C₁₁×C₁₃	Q₈
LCS	G ▷ C₂×C₉×C₁₁×C₁₃ ▷ C₉×C₁₁×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 10296

A minimal faithful permutation representation — 2 generators on 41 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);(1,5,3,7)(2,6,4,8)(10,17)(11,16)(12,15)(13,14)(19,28)(20,27)(21,26)(22,25)(23,24)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)

Polycyclic presentation:

2^3,3,11,13,3|a^2=c,b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^10,f^a=f^12,g^a=g^2

Conjugacy classes & character table

2577 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₁₄₈, Dic₇₇₂₂, Dic₁₂₈₇₀
maximal normal subgroup of	Dic₅₁₄₈
maximal quotient of	Dic₇₇₂₂, Dic₁₂₈₇₀

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, 11, 13, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2574 #72212804