Dic₁₃₇₀₀ #72289902

order	54800
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₇₀₀

Splittings

Hall	C₂₅×C₁₃₇ ⋊ Q₁₆
Sylow	C₁₃₇ ⋊ Dic₁₀₀
Sylow	C₂₅ ⋊ Dic₅₄₈

Subgroups

	subgroup	G/H
center	C₂	D₁₃₇₀₀
commutator, cosocle	C₄×C₂₅×C₁₃₇	C₂²
Frattini	C₄×C₅	C₂×D₆₈₅
Fitting	C₈×C₂₅×C₁₃₇	C₂
socle, prime radical	C₂×C₅×C₁₃₇	D₂₀
hypercenter	C₈	D₃₄₂₅
hypocenter	C₂₅×C₁₃₇	Q₁₆
LCS	G ▷ C₄×C₂₅×C₁₃₇ ▷ C₂×C₂₅×C₁₃₇ ▷ C₂₅×C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 178
transitive degree	μ_t(G) = 54800

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

Polycyclic presentation:

2^4,5,137,5|a^2=d,b^2=cd,c^2=d,e^5=g,b^a=bc,c^a=cd,e^a=e^4g^4,f^a=f^136,g^a=g^4

Conjugacy classes & character table

13703 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, 5, 137, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^136);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13700 #72289902