Dic₁₄₉₅₀ #72292307

order	59800
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₉₅₀

Splittings

Hall	C₁₃×C₂₃×C₂₅ ⋊ Q₈
Hall	C₂₃×C₂₅ ⋊ Dic₂₆
Hall	C₁₃×C₂₅ ⋊ Dic₄₆
Hall	C₁₃×C₂₃ ⋊ Dic₅₀
Sylow	C₂₅ ⋊ Dic₅₉₈
Sylow	C₂₃ ⋊ Dic₆₅₀
Sylow	C₁₃ ⋊ Dic₁₁₅₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₄₇₅
commutator, cosocle	C₂×C₁₃×C₂₃×C₂₅	C₂²
Frattini	C₂×C₅	C₂×D₁₄₉₅
Fitting	C₄×C₁₃×C₂₃×C₂₅	C₂
socle, prime radical	C₂×C₅×C₁₃×C₂₃	C₂×D₅
hypercenter	C₄	D₇₄₇₅
hypocenter	C₁₃×C₂₃×C₂₅	Q₈
LCS	G ▷ C₂×C₁₃×C₂₃×C₂₅ ▷ C₁₃×C₂₃×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 59800

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)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Polycyclic presentation:

2^3,5,13,23,5|a^2=c,b^2=c,d^5=g,b^a=bc,d^a=d^4g^4,e^a=e^12,f^a=f^22,g^a=g^4

Conjugacy classes & character table

14953 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, 13, 23, 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.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14950 #72292307