D₁₂₂₆₇ #37589295

order	24534
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₂₆₇
unambiguous semidirect product	C₉⋊D₁₃₆₃
unambiguous semidirect product	C₂₉⋊D₄₂₃
unambiguous semidirect product	C₄₇⋊D₂₆₁

Splittings

Fitting, derived, Hall	C₉×C₂₉×C₄₇ ⋊ C₂
Hall	C₂₉×C₄₇ ⋊ D₉
Hall	C₉×C₄₇ ⋊ D₂₉
Hall	C₉×C₂₉ ⋊ D₄₇
Sylow	C₄₇ ⋊ D₂₆₁
Sylow	C₂₉ ⋊ D₄₂₃
Sylow	C₉ ⋊ D₁₃₆₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₂₉×C₄₇	C₂
Frattini	C₃	D₄₀₈₉
socle	C₃×C₂₉×C₄₇	D₃
LCS	G ▷ C₉×C₂₉×C₄₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 12267

A minimal faithful permutation representation — 2 generators on 85 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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(2,9)(3,8)(4,7)(5,6)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)

Polycyclic presentation:

2,3,29,47,3|b^3=e^2,b^a=b^2e,c^a=c^28,d^a=d^46,e^a=e^2

Conjugacy classes & character table

6135 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24534, 8);

D12267 #37589295