Dic₁₄₃₆₅ #72291007

order	57460
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₆₅

Splittings

derived, Hall	C₅×C₁₇×C₁₆₉ ⋊ C₄
Hall	C₁₇×C₁₆₉ ⋊ Dic₅
Hall	C₅×C₁₆₉ ⋊ Dic₁₇
Sylow	C₁₆₉ ⋊ Dic₈₅
Hall	C₅×C₁₇ ⋊ Dic₁₆₉
Sylow	C₁₇ ⋊ Dic₈₄₅
Sylow	C₅ ⋊ Dic₂₈₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₃₆₅
commutator, hypocenter	C₅×C₁₇×C₁₆₉	C₄
Frattini	C₂×C₁₃	D₁₁₀₅
Fitting, cosocle	C₂×C₅×C₁₇×C₁₆₉	C₂
socle, prime radical	C₂×C₅×C₁₃×C₁₇	D₁₃
LCS	G ▷ C₅×C₁₇×C₁₆₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 195
transitive degree	μ_t(G) = 57460

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

Polycyclic presentation:

2^2,5,13,17,13|a^2=b,d^13=f,c^a=c^4,d^a=d^12f^12,e^a=e^16,f^a=f^12

Conjugacy classes & character table

14368 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");;
gap> coll := SingleCollector(F, [2, 2, 5, 13, 17, 13]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^12*F.6^12);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14365 #72291007