Dic₁₄₃₇₈ #72291024

order	57512
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₇₈

Splittings

Hall	C₇×C₁₃×C₇₉ ⋊ Q₈
Hall	C₁₃×C₇₉ ⋊ Dic₁₄
Hall	C₇×C₇₉ ⋊ Dic₂₆
Hall	C₇×C₁₃ ⋊ Dic₁₅₈
Sylow	C₇₉ ⋊ Dic₁₈₂
Sylow	C₁₃ ⋊ Dic₁₁₀₆
Sylow	C₇ ⋊ Dic₂₀₅₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₁₈₉
commutator, socle, prime radical, cosocle	C₂×C₇×C₁₃×C₇₉	C₂²
Fitting	C₄×C₇×C₁₃×C₇₉	C₂
hypercenter	C₄	D₇₁₈₉
hypocenter	C₇×C₁₃×C₇₉	Q₈
LCS	G ▷ C₂×C₇×C₁₃×C₇₉ ▷ C₇×C₁₃×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 57512

A minimal faithful permutation representation — 2 generators on 107 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);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2^3,7,13,79|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^12,f^a=f^78

Conjugacy classes & character table

14381 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, 7, 13, 79]);;
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.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^78);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14378 #72291024