Dic₈₆₅₈ #72272995

order	34632
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₆₅₈

Splittings

Hall	C₉×C₁₃×C₃₇ ⋊ Q₈
Hall	C₁₃×C₃₇ ⋊ Dic₁₈
Hall	C₉×C₃₇ ⋊ Dic₂₆
Hall	C₉×C₁₃ ⋊ Dic₇₄
Sylow	C₃₇ ⋊ Dic₂₃₄
Sylow	C₁₃ ⋊ Dic₆₆₆
Sylow	C₉ ⋊ Dic₉₆₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄₃₂₉
commutator, cosocle	C₂×C₉×C₁₃×C₃₇	C₂²
Frattini	C₂×C₃	C₂×D₁₄₄₃
Fitting	C₄×C₉×C₁₃×C₃₇	C₂
socle, prime radical	C₂×C₃×C₁₃×C₃₇	C₂×D₃
hypercenter	C₄	D₄₃₂₉
hypocenter	C₉×C₁₃×C₃₇	Q₈
LCS	G ▷ C₂×C₉×C₁₃×C₃₇ ▷ C₉×C₁₃×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 34632

A minimal faithful permutation representation — 2 generators on 67 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);(1,5,3,7)(2,6,4,8)(10,17)(11,16)(12,15)(13,14)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2^3,3,13,37,3|a^2=c,b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^12,f^a=f^36,g^a=g^2

Conjugacy classes & character table

8661 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, 2, 3, 13, 37, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^36);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic8658 #72272995