D₆₂₀₁ #27137147

order	12402
automorphism	Aut(G) order 23216544
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₆₂₀₁
unambiguous semidirect product	C₉⋊D₆₈₉
unambiguous semidirect product	C₁₃⋊D₄₇₇
unambiguous semidirect product	C₅₃⋊D₁₁₇

Splittings

Fitting, derived, Hall	C₉×C₁₃×C₅₃ ⋊ C₂
Hall	C₁₃×C₅₃ ⋊ D₉
Hall	C₉×C₅₃ ⋊ D₁₃
Hall	C₉×C₁₃ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₁₁₇
Sylow	C₁₃ ⋊ D₄₇₇
Sylow	C₉ ⋊ D₆₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₃×C₅₃	C₂
Frattini	C₃	D₂₀₆₇
socle	C₃×C₁₃×C₅₃	D₃
LCS	G ▷ C₉×C₁₃×C₅₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 6201

A minimal faithful permutation representation — 2 generators on 75 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(2,9)(3,8)(4,7)(5,6)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(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:

2,3,13,53,3|b^3=e^2,b^a=b^2e,c^a=c^12,d^a=d^52,e^a=e^2

Conjugacy classes & character table

3102 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₃×D₁₁₇, D₁₃×D₄₇₇, D₉×D₆₈₉, …
maximal normal subgroup of	D₅₃×D₁₁₇, D₁₃×D₄₇₇, D₉×D₆₈₉, …
maximal quotient of	Dic₆₂₀₁, C₂×D₆₂₀₁, D₁₈₆₀₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 10);

D6201 #27137147