D₁₂₈₄₄ #72235308

order	25688
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₂₈₄₄

Splittings

Fitting	C₄×C₁₉×C₁₆₉ ⋊ C₂
Hall	C₁₉×C₁₆₉ ⋊ D₄
Sylow	C₁₆₉ ⋊ D₇₆
Sylow	C₁₉ ⋊ D₆₇₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃₂₁₁
commutator, cosocle	C₂×C₁₉×C₁₆₉	C₂²
Frattini	C₂×C₁₃	C₂×D₂₄₇
Fitting	C₄×C₁₉×C₁₆₉	C₂
socle	C₂×C₁₃×C₁₉	C₂×D₁₃
hypercenter	C₄	D₃₂₁₁
hypocenter	C₁₉×C₁₆₉	D₄
LCS	G ▷ C₂×C₁₉×C₁₆₉ ▷ C₁₉×C₁₆₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 192
transitive degree	μ_t(G) = 12844

A minimal faithful permutation representation — 2 generators on 192 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)(174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192);(2,4)(6,173)(7,172)(8,171)(9,170)(10,169)(11,168)(12,167)(13,166)(14,165)(15,164)(16,163)(17,162)(18,161)(19,160)(20,159)(21,158)(22,157)(23,156)(24,155)(25,154)(26,153)(27,152)(28,151)(29,150)(30,149)(31,148)(32,147)(33,146)(34,145)(35,144)(36,143)(37,142)(38,141)(39,140)(40,139)(41,138)(42,137)(43,136)(44,135)(45,134)(46,133)(47,132)(48,131)(49,130)(50,129)(51,128)(52,127)(53,126)(54,125)(55,124)(56,123)(57,122)(58,121)(59,120)(60,119)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,112)(68,111)(69,110)(70,109)(71,108)(72,107)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(183,184)

Polycyclic presentation:

2^3,13,19,13|b^2=c,d^13=f,b^a=bc,d^a=d^12f^12,e^a=e^18,f^a=f^12

Conjugacy classes & character table

6425 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₅₆₈₈
maximal normal subgroup of	D₂₅₆₈₈
maximal quotient of	Dic₁₂₈₄₄, D₂₅₆₈₈

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 13, 19, 13]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^12*F.6^12);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D12844 #72235308