Dic₄₂₆₆ #72215657

order	17064
automorphism	Aut(G) order 23957856
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₂₆₆

Splittings

Hall	C₂₇×C₇₉ ⋊ Q₈
Sylow	C₇₉ ⋊ Dic₅₄
Sylow	C₂₇ ⋊ Dic₁₅₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₁₃₃
commutator, cosocle	C₂×C₂₇×C₇₉	C₂²
Frattini	C₂×C₉	C₂×D₂₃₇
Fitting	C₄×C₂₇×C₇₉	C₂
socle, prime radical	C₂×C₃×C₇₉	C₂×D₉
hypercenter	C₄	D₂₁₃₃
hypocenter	C₂₇×C₇₉	Q₈
LCS	G ▷ C₂×C₂₇×C₇₉ ▷ C₂₇×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 114
transitive degree	μ_t(G) = 17064

A minimal faithful permutation representation — 2 generators on 114 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);(1,5,3,7)(2,6,4,8)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2^3,3,79,3^2|a^2=c,b^2=c,d^3=f,f^3=g,b^a=bc,d^a=d^2f^2g^2,e^a=e^78,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

4269 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₅₃₂, Dic₁₂₇₉₈
maximal normal subgroup of	Dic₈₅₃₂
maximal quotient of	Dic₁₂₇₉₈

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 79, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^78);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4266 #72215657