Dic₁₃₆₉₉ #72289899

order	54796
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₆₉₉

Splittings

derived, Hall	C₇×C₁₉×C₁₀₃ ⋊ C₄
Hall	C₁₉×C₁₀₃ ⋊ Dic₇
Hall	C₇×C₁₀₃ ⋊ Dic₁₉
Hall	C₇×C₁₉ ⋊ Dic₁₀₃
Sylow	C₁₀₃ ⋊ Dic₁₃₃
Sylow	C₁₉ ⋊ Dic₇₂₁
Sylow	C₇ ⋊ Dic₁₉₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₆₉₉
commutator, hypocenter	C₇×C₁₉×C₁₀₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₉×C₁₀₃	C₂
LCS	G ▷ C₇×C₁₉×C₁₀₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 54796

A minimal faithful permutation representation — 2 generators on 133 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);(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)(32,133)(33,132)(34,131)(35,130)(36,129)(37,128)(38,127)(39,126)(40,125)(41,124)(42,123)(43,122)(44,121)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)

Polycyclic presentation:

2^2,7,19,103|a^2=b,c^a=c^6,d^a=d^18,e^a=e^102

Conjugacy classes & character table

13702 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, 103]);;
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.5^102);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13699 #72289899