Dic₂₄₇₈ #72212754

order	9912
automorphism	Aut(G) order 6898752
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₄₇₈

Splittings

Hall	C₃×C₇×C₅₉ ⋊ Q₈
Hall	C₇×C₅₉ ⋊ Dic₆
Hall	C₃×C₅₉ ⋊ Dic₁₄
Sylow	C₅₉ ⋊ Dic₄₂
Hall	C₃×C₇ ⋊ Dic₁₁₈
Sylow	C₇ ⋊ Dic₃₅₄
Sylow	C₃ ⋊ Dic₈₂₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₂₃₉
commutator, socle, prime radical, cosocle	C₂×C₃×C₇×C₅₉	C₂²
Fitting	C₃×C₄×C₇×C₅₉	C₂
hypercenter	C₄	D₁₂₃₉
hypocenter	C₃×C₇×C₅₉	Q₈
LCS	G ▷ C₂×C₃×C₇×C₅₉ ▷ C₃×C₇×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 9912

A minimal faithful permutation representation — 2 generators on 77 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);(1,5,3,7)(2,6,4,8)(10,11)(13,18)(14,17)(15,16)(20,77)(21,76)(22,75)(23,74)(24,73)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2^3,3,7,59|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^58

Conjugacy classes & character table

2481 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₄₉₅₆, Dic₇₄₃₄, Dic₁₂₃₉₀
maximal normal subgroup of	Dic₄₉₅₆
maximal quotient of	Dic₇₄₃₄, Dic₁₂₃₉₀

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, 7, 59]);;
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.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2478 #72212754