Dic₁₃₉₇₄ #72290340

order	55896
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₉₇₄

Splittings

Hall	C₃×C₁₇×C₁₃₇ ⋊ Q₈
Hall	C₁₇×C₁₃₇ ⋊ Dic₆
Hall	C₃×C₁₃₇ ⋊ Dic₃₄
Sylow	C₁₃₇ ⋊ Dic₁₀₂
Hall	C₃×C₁₇ ⋊ Dic₂₇₄
Sylow	C₁₇ ⋊ Dic₈₂₂
Sylow	C₃ ⋊ Dic₄₆₅₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₉₈₇
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₇×C₁₃₇	C₂²
Fitting	C₃×C₄×C₁₇×C₁₃₇	C₂
hypercenter	C₄	D₆₉₈₇
hypocenter	C₃×C₁₇×C₁₃₇	Q₈
LCS	G ▷ C₂×C₃×C₁₇×C₁₃₇ ▷ C₃×C₁₇×C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 165
transitive degree	μ_t(G) = 55896

A minimal faithful permutation representation — 2 generators on 165 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);(1,5,3,7)(2,6,4,8)(10,11)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(30,165)(31,164)(32,163)(33,162)(34,161)(35,160)(36,159)(37,158)(38,157)(39,156)(40,155)(41,154)(42,153)(43,152)(44,151)(45,150)(46,149)(47,148)(48,147)(49,146)(50,145)(51,144)(52,143)(53,142)(54,141)(55,140)(56,139)(57,138)(58,137)(59,136)(60,135)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,99)(97,98)

Polycyclic presentation:

2^3,3,17,137|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^16,f^a=f^136

Conjugacy classes & character table

13977 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, 2, 3, 17, 137]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^136);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13974 #72290340