Dic₁₃₄₃₃ #72289321

order	53732
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₄₃₃

Splittings

derived, Hall	C₇×C₁₉×C₁₀₁ ⋊ C₄
Hall	C₁₉×C₁₀₁ ⋊ Dic₇
Hall	C₇×C₁₀₁ ⋊ Dic₁₉
Hall	C₇×C₁₉ ⋊ Dic₁₀₁
Sylow	C₁₀₁ ⋊ Dic₁₃₃
Sylow	C₁₉ ⋊ Dic₇₀₇
Sylow	C₇ ⋊ Dic₁₉₁₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₄₃₃
commutator, hypocenter	C₇×C₁₉×C₁₀₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₉×C₁₀₁	C₂
LCS	G ▷ C₇×C₁₉×C₁₀₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 53732

A minimal faithful permutation representation — 2 generators on 131 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2^2,7,19,101|a^2=b,c^a=c^6,d^a=d^18,e^a=e^100

Conjugacy classes & character table

13436 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, 19, 101]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.5^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13433 #72289321