Dic₁₅₃₈₅ #72293438

order	61540
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₃₈₅

Splittings

derived, Hall	C₅×C₁₇×C₁₈₁ ⋊ C₄
Hall	C₁₇×C₁₈₁ ⋊ Dic₅
Hall	C₅×C₁₈₁ ⋊ Dic₁₇
Sylow	C₁₈₁ ⋊ Dic₈₅
Hall	C₅×C₁₇ ⋊ Dic₁₈₁
Sylow	C₁₇ ⋊ Dic₉₀₅
Sylow	C₅ ⋊ Dic₃₀₇₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₃₈₅
commutator, hypocenter	C₅×C₁₇×C₁₈₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₇×C₁₈₁	C₂
LCS	G ▷ C₅×C₁₇×C₁₈₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 207
transitive degree	μ_t(G) = 61540

A minimal faithful permutation representation — 2 generators on 207 points:

(1,2)(3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207);(1,3,2,4)(6,9)(7,8)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(28,207)(29,206)(30,205)(31,204)(32,203)(33,202)(34,201)(35,200)(36,199)(37,198)(38,197)(39,196)(40,195)(41,194)(42,193)(43,192)(44,191)(45,190)(46,189)(47,188)(48,187)(49,186)(50,185)(51,184)(52,183)(53,182)(54,181)(55,180)(56,179)(57,178)(58,177)(59,176)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,168)(68,167)(69,166)(70,165)(71,164)(72,163)(73,162)(74,161)(75,160)(76,159)(77,158)(78,157)(79,156)(80,155)(81,154)(82,153)(83,152)(84,151)(85,150)(86,149)(87,148)(88,147)(89,146)(90,145)(91,144)(92,143)(93,142)(94,141)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)(101,134)(102,133)(103,132)(104,131)(105,130)(106,129)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)

Polycyclic presentation:

2^2,5,17,181|a^2=b,c^a=c^4,d^a=d^16,e^a=e^180

Conjugacy classes & character table

15388 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");;
gap> coll := SingleCollector(F, [2, 2, 5, 17, 181]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^180);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15385 #72293438