Dic₁₉₅₀ #72212324

order	7800
automorphism	Aut(G) order 3744000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₉₅₀

Splittings

Hall	C₃×C₁₃×C₂₅ ⋊ Q₈
Hall	C₁₃×C₂₅ ⋊ Dic₆
Hall	C₃×C₂₅ ⋊ Dic₂₆
Hall	C₃×C₁₃ ⋊ Dic₅₀
Sylow	C₂₅ ⋊ Dic₇₈
Sylow	C₁₃ ⋊ Dic₁₅₀
Sylow	C₃ ⋊ Dic₆₅₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₉₇₅
commutator, cosocle	C₂×C₃×C₁₃×C₂₅	C₂²
Frattini	C₂×C₅	C₂×D₁₉₅
Fitting	C₃×C₄×C₁₃×C₂₅	C₂
socle, prime radical	C₂×C₃×C₅×C₁₃	C₂×D₅
hypercenter	C₄	D₉₇₅
hypocenter	C₃×C₁₃×C₂₅	Q₈
LCS	G ▷ C₂×C₃×C₁₃×C₂₅ ▷ C₃×C₁₃×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 7800

A minimal faithful permutation representation — 2 generators on 49 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);(1,5,3,7)(2,6,4,8)(10,11)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

1953 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃₉₀₀, Dic₅₈₅₀, Dic₉₇₅₀, …
maximal normal subgroup of	Dic₃₉₀₀
maximal quotient of	Dic₅₈₅₀, Dic₉₇₅₀, Dic₁₃₆₅₀

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

Dic1950 #72212324