Dic₁₄₁₃₀ #72290589

order	56520
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₁₃₀

Splittings

Hall	C₅×C₉×C₁₅₇ ⋊ Q₈
Hall	C₉×C₁₅₇ ⋊ Dic₁₀
Hall	C₅×C₁₅₇ ⋊ Dic₁₈
Sylow	C₁₅₇ ⋊ Dic₉₀
Hall	C₅×C₉ ⋊ Dic₃₁₄
Sylow	C₉ ⋊ Dic₁₅₇₀
Sylow	C₅ ⋊ Dic₂₈₂₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₀₆₅
commutator, cosocle	C₂×C₅×C₉×C₁₅₇	C₂²
Frattini	C₂×C₃	C₂×D₂₃₅₅
Fitting	C₄×C₅×C₉×C₁₅₇	C₂
socle, prime radical	C₂×C₃×C₅×C₁₅₇	C₂×D₃
hypercenter	C₄	D₇₀₆₅
hypocenter	C₅×C₉×C₁₅₇	Q₈
LCS	G ▷ C₂×C₅×C₉×C₁₅₇ ▷ C₅×C₉×C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 179
transitive degree	μ_t(G) = 56520

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

Polycyclic presentation:

2^3,3,5,157,3|a^2=c,b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^4,f^a=f^156,g^a=g^2

Conjugacy classes & character table

14133 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, 3, 5, 157, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^156);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14130 #72290589