Dic₆₄₂₂ #72236094

order	25688
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₄₂₂

Splittings

Hall	C₁₉×C₁₆₉ ⋊ Q₈
Sylow	C₁₆₉ ⋊ Dic₃₈
Sylow	C₁₉ ⋊ Dic₃₃₈

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, prime radical	C₂×C₁₃×C₁₉	C₂×D₁₃
hypercenter	C₄	D₃₂₁₁
hypocenter	C₁₉×C₁₆₉	Q₈
LCS	G ▷ C₂×C₁₉×C₁₆₉ ▷ C₁₉×C₁₆₉
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 196
transitive degree	μ_t(G) = 25688

A minimal faithful permutation representation — 2 generators on 196 points:

(1,2,3,4)(5,8,7,6)(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)(178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196);(1,5,3,7)(2,6,4,8)(10,177)(11,176)(12,175)(13,174)(14,173)(15,172)(16,171)(17,170)(18,169)(19,168)(20,167)(21,166)(22,165)(23,164)(24,163)(25,162)(26,161)(27,160)(28,159)(29,158)(30,157)(31,156)(32,155)(33,154)(34,153)(35,152)(36,151)(37,150)(38,149)(39,148)(40,147)(41,146)(42,145)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,120)(68,119)(69,118)(70,117)(71,116)(72,115)(73,114)(74,113)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(179,196)(180,195)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)

Polycyclic presentation:

2^3,13,19,13|a^2=c,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	Dic₁₂₈₄₄
maximal normal subgroup of	Dic₁₂₈₄₄

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, 1, F.3);;
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);

Dic6422 #72236094