Dic₁₀₁₅₀ #72274237

order	40600
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₁₅₀

Splittings

Hall	C₇×C₂₅×C₂₉ ⋊ Q₈
Hall	C₂₅×C₂₉ ⋊ Dic₁₄
Hall	C₇×C₂₉ ⋊ Dic₅₀
Hall	C₇×C₂₅ ⋊ Dic₅₈
Sylow	C₂₉ ⋊ Dic₃₅₀
Sylow	C₂₅ ⋊ Dic₄₀₆
Sylow	C₇ ⋊ Dic₁₄₅₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₅₀₇₅
commutator, cosocle	C₂×C₇×C₂₅×C₂₉	C₂²
Frattini	C₂×C₅	C₂×D₁₀₁₅
Fitting	C₄×C₇×C₂₅×C₂₉	C₂
socle, prime radical	C₂×C₅×C₇×C₂₉	C₂×D₅
hypercenter	C₄	D₅₀₇₅
hypocenter	C₇×C₂₅×C₂₉	Q₈
LCS	G ▷ C₂×C₇×C₂₅×C₂₉ ▷ C₇×C₂₅×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 40600

A minimal faithful permutation representation — 2 generators on 69 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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69);(1,5,3,7)(2,6,4,8)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(35,40)(36,39)(37,38)(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:

2^3,5,7,29,5|a^2=c,b^2=c,d^5=g,b^a=bc,d^a=d^4g^4,e^a=e^6,f^a=f^28,g^a=g^4

Conjugacy classes & character table

10153 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, 5, 7, 29, 5]);;
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.4^4*F.7^4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic10150 #72274237