D₇₀₁₁ #28380372

order	14022
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₀₁₁
unambiguous semidirect product	C₉⋊D₇₇₉
unambiguous semidirect product	C₁₉⋊D₃₆₉
unambiguous semidirect product	C₄₁⋊D₁₇₁

Splittings

Fitting, derived, Hall	C₉×C₁₉×C₄₁ ⋊ C₂
Hall	C₁₉×C₄₁ ⋊ D₉
Hall	C₉×C₄₁ ⋊ D₁₉
Hall	C₉×C₁₉ ⋊ D₄₁
Sylow	C₄₁ ⋊ D₁₇₁
Sylow	C₁₉ ⋊ D₃₆₉
Sylow	C₉ ⋊ D₇₇₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₉×C₄₁	C₂
Frattini	C₃	D₂₃₃₇
socle	C₃×C₁₉×C₄₁	D₃
LCS	G ▷ C₉×C₁₉×C₄₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 7011

A minimal faithful permutation representation — 2 generators on 69 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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69);(2,9)(3,8)(4,7)(5,6)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(30,69)(31,68)(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:

2,3,19,41,3|b^3=e^2,b^a=b^2e,c^a=c^18,d^a=d^40,e^a=e^2

Conjugacy classes & character table

3507 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₁×D₁₇₁, D₁₉×D₃₆₉, D₉×D₇₇₉, …
maximal normal subgroup of	D₄₁×D₁₇₁, D₁₉×D₃₆₉, D₉×D₇₇₉, …
maximal quotient of	Dic₇₀₁₁, C₂×D₇₀₁₁, D₂₁₀₃₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14022, 9);

D7011 #28380372