Dic₁₃₅₄₂ #72289641

order	54168
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₅₄₂

Splittings

Hall	C₃×C₃₇×C₆₁ ⋊ Q₈
Hall	C₃₇×C₆₁ ⋊ Dic₆
Hall	C₃×C₆₁ ⋊ Dic₇₄
Hall	C₃×C₃₇ ⋊ Dic₁₂₂
Sylow	C₆₁ ⋊ Dic₂₂₂
Sylow	C₃₇ ⋊ Dic₃₆₆
Sylow	C₃ ⋊ Dic₄₅₁₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₇₇₁
commutator, socle, prime radical, cosocle	C₂×C₃×C₃₇×C₆₁	C₂²
Fitting	C₃×C₄×C₃₇×C₆₁	C₂
hypercenter	C₄	D₆₇₇₁
hypocenter	C₃×C₃₇×C₆₁	Q₈
LCS	G ▷ C₂×C₃×C₃₇×C₆₁ ▷ C₃×C₃₇×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 54168

A minimal faithful permutation representation — 2 generators on 109 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);(1,5,3,7)(2,6,4,8)(10,11)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)

Polycyclic presentation:

2^3,3,37,61|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^36,f^a=f^60

Conjugacy classes & character table

13545 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, 37, 61]);;
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.5^36);;
gap> SetConjugate(coll, 6, 1, F.6^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13542 #72289641