Dic₃₂₅₈ #72213617

order	13032
automorphism	Aut(G) order 14074560
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₂₅₈

Splittings

Hall	C₉×C₁₈₁ ⋊ Q₈
Sylow	C₁₈₁ ⋊ Dic₁₈
Sylow	C₉ ⋊ Dic₃₆₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₆₂₉
commutator, cosocle	C₂×C₉×C₁₈₁	C₂²
Frattini	C₂×C₃	C₂×D₅₄₃
Fitting	C₄×C₉×C₁₈₁	C₂
socle, prime radical	C₂×C₃×C₁₈₁	C₂×D₃
hypercenter	C₄	D₁₆₂₉
hypocenter	C₉×C₁₈₁	Q₈
LCS	G ▷ C₂×C₉×C₁₈₁ ▷ C₉×C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 198
transitive degree	μ_t(G) = 13032

A minimal faithful permutation representation — 2 generators on 198 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198);(1,5,3,7)(2,6,4,8)(10,17)(11,16)(12,15)(13,14)(19,198)(20,197)(21,196)(22,195)(23,194)(24,193)(25,192)(26,191)(27,190)(28,189)(29,188)(30,187)(31,186)(32,185)(33,184)(34,183)(35,182)(36,181)(37,180)(38,179)(39,178)(40,177)(41,176)(42,175)(43,174)(44,173)(45,172)(46,171)(47,170)(48,169)(49,168)(50,167)(51,166)(52,165)(53,164)(54,163)(55,162)(56,161)(57,160)(58,159)(59,158)(60,157)(61,156)(62,155)(63,154)(64,153)(65,152)(66,151)(67,150)(68,149)(69,148)(70,147)(71,146)(72,145)(73,144)(74,143)(75,142)(76,141)(77,140)(78,139)(79,138)(80,137)(81,136)(82,135)(83,134)(84,133)(85,132)(86,131)(87,130)(88,129)(89,128)(90,127)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)

Polycyclic presentation:

2^3,3,181,3|a^2=c,b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^180,f^a=f^2

Conjugacy classes & character table

3261 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₅₁₆, Dic₉₇₇₄, Dic₁₆₂₉₀
maximal normal subgroup of	Dic₆₅₁₆
maximal quotient of	Dic₉₇₇₄, Dic₁₆₂₉₀

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, 3, 181, 3]);;
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.4^2*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^180);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3258 #72213617