D₃₁₈₂₅ #72301764

order	63650
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₁₈₂₅

Splittings

Fitting, derived, Hall	C₁₉×C₂₅×C₆₇ ⋊ C₂
Hall	C₂₅×C₆₇ ⋊ D₁₉
Hall	C₁₉×C₆₇ ⋊ D₂₅
Hall	C₁₉×C₂₅ ⋊ D₆₇
Sylow	C₆₇ ⋊ D₄₇₅
Sylow	C₂₅ ⋊ D₁₂₇₃
Sylow	C₁₉ ⋊ D₁₆₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₉×C₂₅×C₆₇	C₂
Frattini	C₅	D₆₃₆₅
socle	C₅×C₁₉×C₆₇	D₅
LCS	G ▷ C₁₉×C₂₅×C₆₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 31825

A minimal faithful permutation representation — 2 generators on 111 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)

Polycyclic presentation:

2,5,19,67,5|b^5=e,b^a=b^4e^4,c^a=c^18,d^a=d^66,e^a=e^4

Conjugacy classes & character table

15914 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");;
gap> coll := SingleCollector(F, [2, 5, 19, 67, 5]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.5^4);;
gap> SetConjugate(coll, 3, 1, F.3^18);;
gap> SetConjugate(coll, 4, 1, F.4^66);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31825 #72301764