D₃₁₅₂₅ #72301469

order	63050
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₁₅₂₅

Splittings

Fitting, derived, Hall	C₁₃×C₂₅×C₉₇ ⋊ C₂
Hall	C₂₅×C₉₇ ⋊ D₁₃
Hall	C₁₃×C₉₇ ⋊ D₂₅
Hall	C₁₃×C₂₅ ⋊ D₉₇
Sylow	C₉₇ ⋊ D₃₂₅
Sylow	C₂₅ ⋊ D₁₂₆₁
Sylow	C₁₃ ⋊ D₂₄₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₃×C₂₅×C₉₇	C₂
Frattini	C₅	D₆₃₀₅
socle	C₅×C₁₃×C₉₇	D₅
LCS	G ▷ C₁₃×C₂₅×C₉₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 31525

A minimal faithful permutation representation — 2 generators on 135 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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,135)(41,134)(42,133)(43,132)(44,131)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,124)(52,123)(53,122)(54,121)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)

Polycyclic presentation:

2,5,13,97,5|b^5=e,b^a=b^4e^4,c^a=c^12,d^a=d^96,e^a=e^4

Conjugacy classes & character table

15764 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, 13, 97, 5]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.5^4);;
gap> SetConjugate(coll, 3, 1, F.3^12);;
gap> SetConjugate(coll, 4, 1, F.4^96);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31525 #72301469