D₉₇₀₉ #34085188

order	19418
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₇₀₉
unambiguous semidirect product	C₇⋊D₁₃₈₇
unambiguous semidirect product	C₁₉⋊D₅₁₁
unambiguous semidirect product	C₇₃⋊D₁₃₃

Splittings

Fitting, socle, derived, Hall	C₇×C₁₉×C₇₃ ⋊ C₂
Hall	C₁₉×C₇₃ ⋊ D₇
Hall	C₇×C₇₃ ⋊ D₁₉
Hall	C₇×C₁₉ ⋊ D₇₃
Sylow	C₇₃ ⋊ D₁₃₃
Sylow	C₁₉ ⋊ D₅₁₁
Sylow	C₇ ⋊ D₁₃₈₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₉×C₇₃	C₂
LCS	G ▷ C₇×C₁₉×C₇₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 9709

A minimal faithful permutation representation — 2 generators on 99 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99);(2,7)(3,6)(4,5)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2,7,19,73|b^a=b^6,c^a=c^18,d^a=d^72

Conjugacy classes & character table

4856 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19418, 8);

D9709 #34085188