Dic₅₉₅₀ #72224244

order	23800
automorphism	Aut(G) order 45696000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₉₅₀

Splittings

Hall	C₇×C₁₇×C₂₅ ⋊ Q₈
Hall	C₁₇×C₂₅ ⋊ Dic₁₄
Hall	C₇×C₂₅ ⋊ Dic₃₄
Hall	C₇×C₁₇ ⋊ Dic₅₀
Sylow	C₂₅ ⋊ Dic₂₃₈
Sylow	C₁₇ ⋊ Dic₃₅₀
Sylow	C₇ ⋊ Dic₈₅₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₉₇₅
commutator, cosocle	C₂×C₇×C₁₇×C₂₅	C₂²
Frattini	C₂×C₅	C₂×D₅₉₅
Fitting	C₄×C₇×C₁₇×C₂₅	C₂
socle, prime radical	C₂×C₅×C₇×C₁₇	C₂×D₅
hypercenter	C₄	D₂₉₇₅
hypocenter	C₇×C₁₇×C₂₅	Q₈
LCS	G ▷ C₂×C₇×C₁₇×C₂₅ ▷ C₇×C₁₇×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 23800

A minimal faithful permutation representation — 2 generators on 57 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);(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)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2^3,5,7,17,5|a^2=c,b^2=c,d^5=g,b^a=bc,d^a=d^4g^4,e^a=e^6,f^a=f^16,g^a=g^4

Conjugacy classes & character table

5953 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₉₀₀
maximal normal subgroup of	Dic₁₁₉₀₀

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, 17, 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.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5950 #72224244