Dic₁₀₀₅₀ #72274147

order	40200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₀₅₀

Splittings

Hall	C₃×C₂₅×C₆₇ ⋊ Q₈
Hall	C₂₅×C₆₇ ⋊ Dic₆
Hall	C₃×C₆₇ ⋊ Dic₅₀
Hall	C₃×C₂₅ ⋊ Dic₁₃₄
Sylow	C₆₇ ⋊ Dic₁₅₀
Sylow	C₂₅ ⋊ Dic₄₀₂
Sylow	C₃ ⋊ Dic₃₃₅₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₅₀₂₅
commutator, cosocle	C₂×C₃×C₂₅×C₆₇	C₂²
Frattini	C₂×C₅	C₂×D₁₀₀₅
Fitting	C₃×C₄×C₂₅×C₆₇	C₂
socle, prime radical	C₂×C₃×C₅×C₆₇	C₂×D₅
hypercenter	C₄	D₅₀₂₅
hypocenter	C₃×C₂₅×C₆₇	Q₈
LCS	G ▷ C₂×C₃×C₂₅×C₆₇ ▷ C₃×C₂₅×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 40200

A minimal faithful permutation representation — 2 generators on 103 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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(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)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2^3,3,5,67,5|a^2=c,b^2=c,e^5=g,b^a=bc,d^a=d^2,e^a=e^4g^4,f^a=f^66,g^a=g^4

Conjugacy classes & character table

10053 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, 3, 5, 67, 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.6^66);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic10050 #72274147