Dic₁₅₄₂₈ #72293518

order	61712
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₄₂₈

Splittings

Hall	C₇×C₁₉×C₂₉ ⋊ Q₁₆
Hall	C₁₉×C₂₉ ⋊ Dic₂₈
Hall	C₇×C₂₉ ⋊ Dic₇₆
Hall	C₇×C₁₉ ⋊ Dic₁₁₆
Sylow	C₂₉ ⋊ Dic₅₃₂
Sylow	C₁₉ ⋊ Dic₈₁₂
Sylow	C₇ ⋊ Dic₂₂₀₄

Subgroups

	subgroup	G/H
center	C₂	D₁₅₄₂₈
commutator, cosocle	C₄×C₇×C₁₉×C₂₉	C₂²
Frattini	C₄	C₂×D₃₈₅₇
Fitting	C₇×C₈×C₁₉×C₂₉	C₂
socle, prime radical	C₂×C₇×C₁₉×C₂₉	D₄
hypercenter	C₈	D₃₈₅₇
hypocenter	C₇×C₁₉×C₂₉	Q₁₆
LCS	G ▷ C₄×C₇×C₁₉×C₂₉ ▷ C₂×C₇×C₁₉×C₂₉ ▷ C₇×C₁₉×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 61712

A minimal faithful permutation representation — 2 generators on 71 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,23)(19,22)(20,21)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2^4,7,19,29|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^18,g^a=g^28

Conjugacy classes & character table

15431 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, 19, 29]);;
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.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15428 #72293518