Dic₁₂₉₇₁ #72288481

order	51884
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₉₇₁

Splittings

derived, Hall	C₇×C₁₇×C₁₀₉ ⋊ C₄
Hall	C₁₇×C₁₀₉ ⋊ Dic₇
Hall	C₇×C₁₀₉ ⋊ Dic₁₇
Hall	C₇×C₁₇ ⋊ Dic₁₀₉
Sylow	C₁₀₉ ⋊ Dic₁₁₉
Sylow	C₁₇ ⋊ Dic₇₆₃
Sylow	C₇ ⋊ Dic₁₈₅₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₂₉₇₁
commutator, hypocenter	C₇×C₁₇×C₁₀₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₇×C₁₀₉	C₂
LCS	G ▷ C₇×C₁₇×C₁₀₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 137
transitive degree	μ_t(G) = 51884

A minimal faithful permutation representation — 2 generators on 137 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(30,137)(31,136)(32,135)(33,134)(34,133)(35,132)(36,131)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,122)(46,121)(47,120)(48,119)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)

Polycyclic presentation:

2^2,7,17,109|a^2=b,c^a=c^6,d^a=d^16,e^a=e^108

Conjugacy classes & character table

12974 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, 7, 17, 109]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^108);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12971 #72288481