Dic₆₂₇₈ #72236047

order	25112
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₂₇₈

Splittings

Hall	C₄₃×C₇₃ ⋊ Q₈
Sylow	C₇₃ ⋊ Dic₈₆
Sylow	C₄₃ ⋊ Dic₁₄₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₁₃₉
commutator, socle, prime radical, cosocle	C₂×C₄₃×C₇₃	C₂²
Fitting	C₄×C₄₃×C₇₃	C₂
hypercenter	C₄	D₃₁₃₉
hypocenter	C₄₃×C₇₃	Q₈
LCS	G ▷ C₂×C₄₃×C₇₃ ▷ C₄₃×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 124
transitive degree	μ_t(G) = 25112

A minimal faithful permutation representation — 2 generators on 124 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)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124);(1,5,3,7)(2,6,4,8)(10,51)(11,50)(12,49)(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)(53,124)(54,123)(55,122)(56,121)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)

Polycyclic presentation:

2^3,43,73|a^2=c,b^2=c,b^a=bc,d^a=d^42,e^a=e^72

Conjugacy classes & character table

6281 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₅₅₆
maximal normal subgroup of	Dic₁₂₅₅₆

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, 2, 43, 73]);;
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.4^42);;
gap> SetConjugate(coll, 5, 1, F.5^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6278 #72236047