Dic₆₃₁₄ #72236055

order	25256
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₃₁₄

Splittings

Hall	C₇×C₁₁×C₄₁ ⋊ Q₈
Hall	C₁₁×C₄₁ ⋊ Dic₁₄
Hall	C₇×C₄₁ ⋊ Dic₂₂
Hall	C₇×C₁₁ ⋊ Dic₈₂
Sylow	C₄₁ ⋊ Dic₁₅₄
Sylow	C₁₁ ⋊ Dic₅₇₄
Sylow	C₇ ⋊ Dic₉₀₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₁₅₇
commutator, socle, prime radical, cosocle	C₂×C₇×C₁₁×C₄₁	C₂²
Fitting	C₄×C₇×C₁₁×C₄₁	C₂
hypercenter	C₄	D₃₁₅₇
hypocenter	C₇×C₁₁×C₄₁	Q₈
LCS	G ▷ C₂×C₇×C₁₁×C₄₁ ▷ C₇×C₁₁×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 25256

A minimal faithful permutation representation — 2 generators on 67 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);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,26)(18,25)(19,24)(20,23)(21,22)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2^3,7,11,41|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^10,f^a=f^40

Conjugacy classes & character table

6317 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₆₂₈
maximal normal subgroup of	Dic₁₂₆₂₈

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, 7, 11, 41]);;
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.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6314 #72236055