Dic₁₀₇₄₀ #72274792

order	42960
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₇₄₀

Splittings

Hall	C₃×C₅×C₁₇₉ ⋊ Q₁₆
Hall	C₅×C₁₇₉ ⋊ Dic₁₂
Hall	C₃×C₁₇₉ ⋊ Dic₂₀
Sylow	C₁₇₉ ⋊ Dic₆₀
Hall	C₃×C₅ ⋊ Dic₇₁₆
Sylow	C₅ ⋊ Dic₂₁₄₈
Sylow	C₃ ⋊ Dic₃₅₈₀

Subgroups

	subgroup	G/H
center	C₂	D₁₀₇₄₀
commutator, cosocle	C₃×C₄×C₅×C₁₇₉	C₂²
Frattini	C₄	C₂×D₂₆₈₅
Fitting	C₃×C₅×C₈×C₁₇₉	C₂
socle, prime radical	C₂×C₃×C₅×C₁₇₉	D₄
hypercenter	C₈	D₂₆₈₅
hypocenter	C₃×C₅×C₁₇₉	Q₁₆
LCS	G ▷ C₃×C₄×C₅×C₁₇₉ ▷ C₂×C₃×C₅×C₁₇₉ ▷ C₃×C₅×C₁₇₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 203
transitive degree	μ_t(G) = 42960

A minimal faithful permutation representation — 2 generators on 203 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19)(20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,24)(22,23)(26,203)(27,202)(28,201)(29,200)(30,199)(31,198)(32,197)(33,196)(34,195)(35,194)(36,193)(37,192)(38,191)(39,190)(40,189)(41,188)(42,187)(43,186)(44,185)(45,184)(46,183)(47,182)(48,181)(49,180)(50,179)(51,178)(52,177)(53,176)(54,175)(55,174)(56,173)(57,172)(58,171)(59,170)(60,169)(61,168)(62,167)(63,166)(64,165)(65,164)(66,163)(67,162)(68,161)(69,160)(70,159)(71,158)(72,157)(73,156)(74,155)(75,154)(76,153)(77,152)(78,151)(79,150)(80,149)(81,148)(82,147)(83,146)(84,145)(85,144)(86,143)(87,142)(88,141)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)

Polycyclic presentation:

2^4,3,5,179|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^178

Conjugacy classes & character table

10743 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, 3, 5, 179]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^178);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic10740 #72274792