D₂₁₉₂₄ #72275336

order	43848
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₁₉₂₄

Splittings

Fitting	C₄×C₇×C₂₇×C₂₉ ⋊ C₂
Hall	C₇×C₂₇×C₂₉ ⋊ D₄
Hall	C₂₇×C₂₉ ⋊ D₂₈
Hall	C₇×C₂₉ ⋊ D₁₀₈
Hall	C₇×C₂₇ ⋊ D₁₁₆
Sylow	C₂₉ ⋊ D₇₅₆
Sylow	C₂₇ ⋊ D₈₁₂
Sylow	C₇ ⋊ D₃₁₃₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₅₄₈₁
commutator, cosocle	C₂×C₇×C₂₇×C₂₉	C₂²
Frattini	C₂×C₉	C₂×D₆₀₉
Fitting	C₄×C₇×C₂₇×C₂₉	C₂
socle	C₂×C₃×C₇×C₂₉	C₂×D₉
hypercenter	C₄	D₅₄₈₁
hypocenter	C₇×C₂₇×C₂₉	D₄
LCS	G ▷ C₂×C₇×C₂₇×C₂₉ ▷ C₇×C₂₇×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 21924

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)(33,38)(34,37)(35,36)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2^3,3,7,29,3^2|b^2=c,d^3=g,g^3=h,b^a=bc,d^a=d^2g^2h^2,e^a=e^6,f^a=f^28,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

10965 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, 7, 29, 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.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
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);

D21924 #72275336