Dic₁₃₄₆₈ #72289413

order	53872
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₄₆₈

Splittings

Hall	C₇×C₁₃×C₃₇ ⋊ Q₁₆
Hall	C₁₃×C₃₇ ⋊ Dic₂₈
Hall	C₇×C₃₇ ⋊ Dic₅₂
Hall	C₇×C₁₃ ⋊ Dic₁₄₈
Sylow	C₃₇ ⋊ Dic₃₆₄
Sylow	C₁₃ ⋊ Dic₁₀₃₆
Sylow	C₇ ⋊ Dic₁₉₂₄

Subgroups

	subgroup	G/H
center	C₂	D₁₃₄₆₈
commutator, cosocle	C₄×C₇×C₁₃×C₃₇	C₂²
Frattini	C₄	C₂×D₃₃₆₇
Fitting	C₇×C₈×C₁₃×C₃₇	C₂
socle, prime radical	C₂×C₇×C₁₃×C₃₇	D₄
hypercenter	C₈	D₃₃₆₇
hypocenter	C₇×C₁₃×C₃₇	Q₁₆
LCS	G ▷ C₄×C₇×C₁₃×C₃₇ ▷ C₂×C₇×C₁₃×C₃₇ ▷ C₇×C₁₃×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 53872

A minimal faithful permutation representation — 2 generators on 73 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,23)(19,22)(20,21)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2^4,7,13,37|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^12,g^a=g^36

Conjugacy classes & character table

13471 conjugacy classes. The character table is too large to display.

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, 2, 7, 13, 37]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13468 #72289413