D₃₂₂₉₂ #72302008

order	64584
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₂₉₂

Splittings

Fitting	C₄×C₁₃×C₂₃×C₂₇ ⋊ C₂
Hall	C₁₃×C₂₃×C₂₇ ⋊ D₄
Hall	C₂₃×C₂₇ ⋊ D₅₂
Hall	C₁₃×C₂₇ ⋊ D₉₂
Hall	C₁₃×C₂₃ ⋊ D₁₀₈
Sylow	C₂₇ ⋊ D₁₁₉₆
Sylow	C₂₃ ⋊ D₁₄₀₄
Sylow	C₁₃ ⋊ D₂₄₈₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₈₀₇₃
commutator, cosocle	C₂×C₁₃×C₂₃×C₂₇	C₂²
Frattini	C₂×C₉	C₂×D₈₉₇
Fitting	C₄×C₁₃×C₂₃×C₂₇	C₂
socle	C₂×C₃×C₁₃×C₂₃	C₂×D₉
hypercenter	C₄	D₈₀₇₃
hypocenter	C₁₃×C₂₃×C₂₇	D₄
LCS	G ▷ C₂×C₁₃×C₂₃×C₂₇ ▷ C₁₃×C₂₃×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 32292

A minimal faithful permutation representation — 2 generators on 67 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);(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)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2^3,3,13,23,3^2|b^2=c,d^3=g,g^3=h,b^a=bc,d^a=d^2g^2h^2,e^a=e^12,f^a=f^22,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

16149 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 13, 23, 3, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32292 #72302008