D₆₁₄₉ #27134136

order	12298
automorphism	Aut(G) order 30990960
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₆₁₄₉
unambiguous semidirect product	C₁₁⋊D₅₅₉
unambiguous semidirect product	C₁₃⋊D₄₇₃
unambiguous semidirect product	C₄₃⋊D₁₄₃

Splittings

Fitting, socle, derived, Hall	C₁₁×C₁₃×C₄₃ ⋊ C₂
Hall	C₁₃×C₄₃ ⋊ D₁₁
Hall	C₁₁×C₄₃ ⋊ D₁₃
Hall	C₁₁×C₁₃ ⋊ D₄₃
Sylow	C₄₃ ⋊ D₁₄₃
Sylow	C₁₃ ⋊ D₄₇₃
Sylow	C₁₁ ⋊ D₅₅₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁×C₁₃×C₄₃	C₂
LCS	G ▷ C₁₁×C₁₃×C₄₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 6149

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);(2,11)(3,10)(4,9)(5,8)(6,7)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2,11,13,43|b^a=b^10,c^a=c^12,d^a=d^42

Conjugacy classes & character table

3076 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₃×D₁₄₃, D₁₃×D₄₇₃, D₁₁×D₅₅₉, …
maximal normal subgroup of	D₄₃×D₁₄₃, D₁₃×D₄₇₃, D₁₁×D₅₅₉, …
maximal quotient of	Dic₆₁₄₉, C₂×D₆₁₄₉, C₃×D₆₁₄₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12298, 8);

D6149 #27134136