D₂₆₂₂₄ #72289582

order	52448
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₂₂₄

Splittings

Fitting	C₁₁×C₁₆×C₁₄₉ ⋊ C₂
Hall	C₁₁×C₁₄₉ ⋊ D₁₆
Sylow	C₁₄₉ ⋊ D₁₇₆
Sylow	C₁₁ ⋊ D₂₃₈₄

Subgroups

	subgroup	G/H
center	C₂	D₁₃₁₁₂
commutator, cosocle	C₈×C₁₁×C₁₄₉	C₂²
Frattini	C₈	C₂×D₁₆₃₉
Fitting	C₁₁×C₁₆×C₁₄₉	C₂
socle	C₂×C₁₁×C₁₄₉	D₈
hypercenter	C₁₆	D₁₆₃₉
hypocenter	C₁₁×C₁₄₉	D₁₆
LCS	G ▷ C₈×C₁₁×C₁₄₉ ▷ C₄×C₁₁×C₁₄₉ ▷ C₂×C₁₁×C₁₄₉ ▷ C₁₁×C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 176
transitive degree	μ_t(G) = 26224

A minimal faithful permutation representation — 2 generators on 176 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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,27)(19,26)(20,25)(21,24)(22,23)(29,176)(30,175)(31,174)(32,173)(33,172)(34,171)(35,170)(36,169)(37,168)(38,167)(39,166)(40,165)(41,164)(42,163)(43,162)(44,161)(45,160)(46,159)(47,158)(48,157)(49,156)(50,155)(51,154)(52,153)(53,152)(54,151)(55,150)(56,149)(57,148)(58,147)(59,146)(60,145)(61,144)(62,143)(63,142)(64,141)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(71,134)(72,133)(73,132)(74,131)(75,130)(76,129)(77,128)(78,127)(79,126)(80,125)(81,124)(82,123)(83,122)(84,121)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)

Polycyclic presentation:

2^5,11,149|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^10,g^a=g^148

Conjugacy classes & character table

13115 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, 11, 149]);;
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.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^148);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26224 #72289582