D₂₆₃₈₄ #72289736

order	52768
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₃₈₄

Splittings

Fitting	C₁₆×C₁₇×C₉₇ ⋊ C₂
Hall	C₁₇×C₉₇ ⋊ D₁₆
Sylow	C₉₇ ⋊ D₂₇₂
Sylow	C₁₇ ⋊ D₁₅₅₂

Subgroups

	subgroup	G/H
center	C₂	D₁₃₁₉₂
commutator, cosocle	C₈×C₁₇×C₉₇	C₂²
Frattini	C₈	C₂×D₁₆₄₉
Fitting	C₁₆×C₁₇×C₉₇	C₂
socle	C₂×C₁₇×C₉₇	D₈
hypercenter	C₁₆	D₁₆₄₉
hypocenter	C₁₇×C₉₇	D₁₆
LCS	G ▷ C₈×C₁₇×C₉₇ ▷ C₄×C₁₇×C₉₇ ▷ C₂×C₁₇×C₉₇ ▷ C₁₇×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 130
transitive degree	μ_t(G) = 26384

A minimal faithful permutation representation — 2 generators on 130 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(35,130)(36,129)(37,128)(38,127)(39,126)(40,125)(41,124)(42,123)(43,122)(44,121)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)

Polycyclic presentation:

2^5,17,97|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^16,g^a=g^96

Conjugacy classes & character table

13195 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, 2, 2, 17, 97]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26384 #72289736