Dic₁₄₉₃₁ #72292285

order	59724
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₉₃₁

Splittings

derived, Hall	C₇×C₂₇×C₇₉ ⋊ C₄
Hall	C₂₇×C₇₉ ⋊ Dic₇
Hall	C₇×C₇₉ ⋊ Dic₂₇
Hall	C₇×C₂₇ ⋊ Dic₇₉
Sylow	C₇₉ ⋊ Dic₁₈₉
Sylow	C₂₇ ⋊ Dic₅₅₃
Sylow	C₇ ⋊ Dic₂₁₃₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₉₃₁
commutator, hypocenter	C₇×C₂₇×C₇₉	C₄
Frattini	C₂×C₉	D₁₆₅₉
Fitting, cosocle	C₂×C₇×C₂₇×C₇₉	C₂
socle, prime radical	C₂×C₃×C₇×C₇₉	D₉
LCS	G ▷ C₇×C₂₇×C₇₉
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 59724

A minimal faithful permutation representation — 2 generators on 117 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);(1,3,2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,38)(34,37)(35,36)(40,117)(41,116)(42,115)(43,114)(44,113)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)

Polycyclic presentation:

2^2,3,7,79,3^2|a^2=b,c^3=f,f^3=g,c^a=c^2f^2g^2,d^a=d^6,e^a=e^78,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

14934 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 3, 7, 79, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
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);

Dic14931 #72292285