Dic₁₄₈₇₄ #72292192

order	59496
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₈₇₄

Splittings

Hall	C₃×C₃₇×C₆₇ ⋊ Q₈
Hall	C₃₇×C₆₇ ⋊ Dic₆
Hall	C₃×C₆₇ ⋊ Dic₇₄
Hall	C₃×C₃₇ ⋊ Dic₁₃₄
Sylow	C₆₇ ⋊ Dic₂₂₂
Sylow	C₃₇ ⋊ Dic₄₀₂
Sylow	C₃ ⋊ Dic₄₉₅₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₄₃₇
commutator, socle, prime radical, cosocle	C₂×C₃×C₃₇×C₆₇	C₂²
Fitting	C₃×C₄×C₃₇×C₆₇	C₂
hypercenter	C₄	D₇₄₃₇
hypocenter	C₃×C₃₇×C₆₇	Q₈
LCS	G ▷ C₂×C₃×C₃₇×C₆₇ ▷ C₃×C₃₇×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 59496

A minimal faithful permutation representation — 2 generators on 115 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);(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)(50,115)(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:

2^3,3,37,67|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^36,f^a=f^66

Conjugacy classes & character table

14877 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, 67]);;
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.6^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14874 #72292192