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#4296

order	171
isoclinism	isoclinic class of #4296
automorphism	Aut(G) = AGL(1,𝔽₁₉) (order 342) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72244912
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow

C₁₉ ⋊ C₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₁₉	C₉
prime radical, cosocle	C₁₉⋊C₃	C₃
LCS	G ▷ C₁₉
UCS	C₁

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 19

A minimal faithful permutation representation — 2 generators on 19 points:

(2,3,6,4,7,13,9,14,8)(5,10,17,11,18,19,12,16,15);(1,2,5,12,3,8,14,4,10,7,15,9,16,17,11,19,6,13,18)

Polycyclic presentation:

3^2,19|a^3=b,c^a=c^4,c^b=c^7

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	9	9	19	19	19	19	19	19	19	19
order	1	19	19	3	3	9	9	9	9	9	9
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	ζ⁴+ζ⁷+ζ¹³+ζ¹⁶+ζ²²+ζ²⁵+ζ³¹+ζ³⁴+ζ⁴⁰+ζ⁴³+ζ⁴⁹+ζ⁵²+ζ⁵⁸+ζ⁶¹+ζ⁶⁷+ζ⁷⁰+ζ⁷⁹+ζ⁸⁵+ζ⁸⁸+ζ⁹⁴+ζ⁹⁷+ζ¹⁰³+ζ¹⁰⁶+ζ¹¹²+ζ¹¹⁵+ζ¹²¹+ζ¹²⁴+ζ¹³⁰+ζ¹³⁹+ζ¹⁴²+ζ¹⁴⁸+ζ¹⁵¹+ζ¹⁵⁷+ζ¹⁶⁰+ζ¹⁶⁶+ζ¹⁶⁹	−ζ⁴−ζ¹³−ζ²²−ζ³¹−ζ⁴⁰−ζ⁴⁹−ζ⁵⁸−ζ⁶⁷−ζ⁸⁵−ζ⁹⁴−ζ¹⁰³−ζ¹¹²−ζ¹²¹−ζ¹³⁰−ζ¹³⁹−ζ¹⁴⁸−ζ¹⁵⁷−ζ¹⁶⁶	−ζ⁷−ζ¹⁶−ζ²⁵−ζ³⁴−ζ⁴³−ζ⁵²−ζ⁶¹−ζ⁷⁰−ζ⁷⁹−ζ⁸⁸−ζ⁹⁷−ζ¹⁰⁶−ζ¹¹⁵−ζ¹²⁴−ζ¹⁴²−ζ¹⁵¹−ζ¹⁶⁰−ζ¹⁶⁹	ζ⁵+ζ⁸+ζ¹⁴+ζ¹⁷+ζ²³+ζ²⁶+ζ³²+ζ³⁵+ζ⁴¹+ζ⁴⁴+ζ⁵⁰+ζ⁵³+ζ⁵⁹+ζ⁶²+ζ⁶⁸+ζ⁷¹+ζ⁷⁷+ζ⁸⁰+ζ⁸⁶+ζ⁸⁹+ζ⁹⁸+ζ¹⁰⁴+ζ¹⁰⁷+ζ¹¹³+ζ¹¹⁶+ζ¹²²+ζ¹²⁵+ζ¹³¹+ζ¹³⁴+ζ¹⁴⁰+ζ¹⁴³+ζ¹⁴⁹+ζ¹⁵⁸+ζ¹⁶¹+ζ¹⁶⁷+ζ¹⁷⁰	−ζ⁵−ζ¹⁴−ζ²³−ζ³²−ζ⁴¹−ζ⁵⁰−ζ⁵⁹−ζ⁶⁸−ζ⁷⁷−ζ⁸⁶−ζ¹⁰⁴−ζ¹¹³−ζ¹²²−ζ¹³¹−ζ¹⁴⁰−ζ¹⁴⁹−ζ¹⁵⁸−ζ¹⁶⁷	−ζ⁸−ζ¹⁷−ζ²⁶−ζ³⁵−ζ⁴⁴−ζ⁵³−ζ⁶²−ζ⁷¹−ζ⁸⁰−ζ⁸⁹−ζ⁹⁸−ζ¹⁰⁷−ζ¹¹⁶−ζ¹²⁵−ζ¹³⁴−ζ¹⁴³−ζ¹⁶¹−ζ¹⁷⁰
χ₅	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	−ζ⁴−ζ¹³−ζ²²−ζ³¹−ζ⁴⁰−ζ⁴⁹−ζ⁵⁸−ζ⁶⁷−ζ⁸⁵−ζ⁹⁴−ζ¹⁰³−ζ¹¹²−ζ¹²¹−ζ¹³⁰−ζ¹³⁹−ζ¹⁴⁸−ζ¹⁵⁷−ζ¹⁶⁶	−ζ⁷−ζ¹⁶−ζ²⁵−ζ³⁴−ζ⁴³−ζ⁵²−ζ⁶¹−ζ⁷⁰−ζ⁷⁹−ζ⁸⁸−ζ⁹⁷−ζ¹⁰⁶−ζ¹¹⁵−ζ¹²⁴−ζ¹⁴²−ζ¹⁵¹−ζ¹⁶⁰−ζ¹⁶⁹	ζ⁴+ζ⁷+ζ¹³+ζ¹⁶+ζ²²+ζ²⁵+ζ³¹+ζ³⁴+ζ⁴⁰+ζ⁴³+ζ⁴⁹+ζ⁵²+ζ⁵⁸+ζ⁶¹+ζ⁶⁷+ζ⁷⁰+ζ⁷⁹+ζ⁸⁵+ζ⁸⁸+ζ⁹⁴+ζ⁹⁷+ζ¹⁰³+ζ¹⁰⁶+ζ¹¹²+ζ¹¹⁵+ζ¹²¹+ζ¹²⁴+ζ¹³⁰+ζ¹³⁹+ζ¹⁴²+ζ¹⁴⁸+ζ¹⁵¹+ζ¹⁵⁷+ζ¹⁶⁰+ζ¹⁶⁶+ζ¹⁶⁹	−ζ⁸−ζ¹⁷−ζ²⁶−ζ³⁵−ζ⁴⁴−ζ⁵³−ζ⁶²−ζ⁷¹−ζ⁸⁰−ζ⁸⁹−ζ⁹⁸−ζ¹⁰⁷−ζ¹¹⁶−ζ¹²⁵−ζ¹³⁴−ζ¹⁴³−ζ¹⁶¹−ζ¹⁷⁰	ζ⁵+ζ⁸+ζ¹⁴+ζ¹⁷+ζ²³+ζ²⁶+ζ³²+ζ³⁵+ζ⁴¹+ζ⁴⁴+ζ⁵⁰+ζ⁵³+ζ⁵⁹+ζ⁶²+ζ⁶⁸+ζ⁷¹+ζ⁷⁷+ζ⁸⁰+ζ⁸⁶+ζ⁸⁹+ζ⁹⁸+ζ¹⁰⁴+ζ¹⁰⁷+ζ¹¹³+ζ¹¹⁶+ζ¹²²+ζ¹²⁵+ζ¹³¹+ζ¹³⁴+ζ¹⁴⁰+ζ¹⁴³+ζ¹⁴⁹+ζ¹⁵⁸+ζ¹⁶¹+ζ¹⁶⁷+ζ¹⁷⁰	−ζ⁵−ζ¹⁴−ζ²³−ζ³²−ζ⁴¹−ζ⁵⁰−ζ⁵⁹−ζ⁶⁸−ζ⁷⁷−ζ⁸⁶−ζ¹⁰⁴−ζ¹¹³−ζ¹²²−ζ¹³¹−ζ¹⁴⁰−ζ¹⁴⁹−ζ¹⁵⁸−ζ¹⁶⁷
χ₆	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	−ζ⁷−ζ¹⁶−ζ²⁵−ζ³⁴−ζ⁴³−ζ⁵²−ζ⁶¹−ζ⁷⁰−ζ⁷⁹−ζ⁸⁸−ζ⁹⁷−ζ¹⁰⁶−ζ¹¹⁵−ζ¹²⁴−ζ¹⁴²−ζ¹⁵¹−ζ¹⁶⁰−ζ¹⁶⁹	ζ⁴+ζ⁷+ζ¹³+ζ¹⁶+ζ²²+ζ²⁵+ζ³¹+ζ³⁴+ζ⁴⁰+ζ⁴³+ζ⁴⁹+ζ⁵²+ζ⁵⁸+ζ⁶¹+ζ⁶⁷+ζ⁷⁰+ζ⁷⁹+ζ⁸⁵+ζ⁸⁸+ζ⁹⁴+ζ⁹⁷+ζ¹⁰³+ζ¹⁰⁶+ζ¹¹²+ζ¹¹⁵+ζ¹²¹+ζ¹²⁴+ζ¹³⁰+ζ¹³⁹+ζ¹⁴²+ζ¹⁴⁸+ζ¹⁵¹+ζ¹⁵⁷+ζ¹⁶⁰+ζ¹⁶⁶+ζ¹⁶⁹	−ζ⁴−ζ¹³−ζ²²−ζ³¹−ζ⁴⁰−ζ⁴⁹−ζ⁵⁸−ζ⁶⁷−ζ⁸⁵−ζ⁹⁴−ζ¹⁰³−ζ¹¹²−ζ¹²¹−ζ¹³⁰−ζ¹³⁹−ζ¹⁴⁸−ζ¹⁵⁷−ζ¹⁶⁶	−ζ⁵−ζ¹⁴−ζ²³−ζ³²−ζ⁴¹−ζ⁵⁰−ζ⁵⁹−ζ⁶⁸−ζ⁷⁷−ζ⁸⁶−ζ¹⁰⁴−ζ¹¹³−ζ¹²²−ζ¹³¹−ζ¹⁴⁰−ζ¹⁴⁹−ζ¹⁵⁸−ζ¹⁶⁷	−ζ⁸−ζ¹⁷−ζ²⁶−ζ³⁵−ζ⁴⁴−ζ⁵³−ζ⁶²−ζ⁷¹−ζ⁸⁰−ζ⁸⁹−ζ⁹⁸−ζ¹⁰⁷−ζ¹¹⁶−ζ¹²⁵−ζ¹³⁴−ζ¹⁴³−ζ¹⁶¹−ζ¹⁷⁰	ζ⁵+ζ⁸+ζ¹⁴+ζ¹⁷+ζ²³+ζ²⁶+ζ³²+ζ³⁵+ζ⁴¹+ζ⁴⁴+ζ⁵⁰+ζ⁵³+ζ⁵⁹+ζ⁶²+ζ⁶⁸+ζ⁷¹+ζ⁷⁷+ζ⁸⁰+ζ⁸⁶+ζ⁸⁹+ζ⁹⁸+ζ¹⁰⁴+ζ¹⁰⁷+ζ¹¹³+ζ¹¹⁶+ζ¹²²+ζ¹²⁵+ζ¹³¹+ζ¹³⁴+ζ¹⁴⁰+ζ¹⁴³+ζ¹⁴⁹+ζ¹⁵⁸+ζ¹⁶¹+ζ¹⁶⁷+ζ¹⁷⁰
χ₇	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	ζ⁵+ζ⁸+ζ¹⁴+ζ¹⁷+ζ²³+ζ²⁶+ζ³²+ζ³⁵+ζ⁴¹+ζ⁴⁴+ζ⁵⁰+ζ⁵³+ζ⁵⁹+ζ⁶²+ζ⁶⁸+ζ⁷¹+ζ⁷⁷+ζ⁸⁰+ζ⁸⁶+ζ⁸⁹+ζ⁹⁸+ζ¹⁰⁴+ζ¹⁰⁷+ζ¹¹³+ζ¹¹⁶+ζ¹²²+ζ¹²⁵+ζ¹³¹+ζ¹³⁴+ζ¹⁴⁰+ζ¹⁴³+ζ¹⁴⁹+ζ¹⁵⁸+ζ¹⁶¹+ζ¹⁶⁷+ζ¹⁷⁰	−ζ⁸−ζ¹⁷−ζ²⁶−ζ³⁵−ζ⁴⁴−ζ⁵³−ζ⁶²−ζ⁷¹−ζ⁸⁰−ζ⁸⁹−ζ⁹⁸−ζ¹⁰⁷−ζ¹¹⁶−ζ¹²⁵−ζ¹³⁴−ζ¹⁴³−ζ¹⁶¹−ζ¹⁷⁰	−ζ⁵−ζ¹⁴−ζ²³−ζ³²−ζ⁴¹−ζ⁵⁰−ζ⁵⁹−ζ⁶⁸−ζ⁷⁷−ζ⁸⁶−ζ¹⁰⁴−ζ¹¹³−ζ¹²²−ζ¹³¹−ζ¹⁴⁰−ζ¹⁴⁹−ζ¹⁵⁸−ζ¹⁶⁷	−ζ⁴−ζ¹³−ζ²²−ζ³¹−ζ⁴⁰−ζ⁴⁹−ζ⁵⁸−ζ⁶⁷−ζ⁸⁵−ζ⁹⁴−ζ¹⁰³−ζ¹¹²−ζ¹²¹−ζ¹³⁰−ζ¹³⁹−ζ¹⁴⁸−ζ¹⁵⁷−ζ¹⁶⁶	ζ⁴+ζ⁷+ζ¹³+ζ¹⁶+ζ²²+ζ²⁵+ζ³¹+ζ³⁴+ζ⁴⁰+ζ⁴³+ζ⁴⁹+ζ⁵²+ζ⁵⁸+ζ⁶¹+ζ⁶⁷+ζ⁷⁰+ζ⁷⁹+ζ⁸⁵+ζ⁸⁸+ζ⁹⁴+ζ⁹⁷+ζ¹⁰³+ζ¹⁰⁶+ζ¹¹²+ζ¹¹⁵+ζ¹²¹+ζ¹²⁴+ζ¹³⁰+ζ¹³⁹+ζ¹⁴²+ζ¹⁴⁸+ζ¹⁵¹+ζ¹⁵⁷+ζ¹⁶⁰+ζ¹⁶⁶+ζ¹⁶⁹	−ζ⁷−ζ¹⁶−ζ²⁵−ζ³⁴−ζ⁴³−ζ⁵²−ζ⁶¹−ζ⁷⁰−ζ⁷⁹−ζ⁸⁸−ζ⁹⁷−ζ¹⁰⁶−ζ¹¹⁵−ζ¹²⁴−ζ¹⁴²−ζ¹⁵¹−ζ¹⁶⁰−ζ¹⁶⁹
χ₈	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	−ζ⁵−ζ¹⁴−ζ²³−ζ³²−ζ⁴¹−ζ⁵⁰−ζ⁵⁹−ζ⁶⁸−ζ⁷⁷−ζ⁸⁶−ζ¹⁰⁴−ζ¹¹³−ζ¹²²−ζ¹³¹−ζ¹⁴⁰−ζ¹⁴⁹−ζ¹⁵⁸−ζ¹⁶⁷	ζ⁵+ζ⁸+ζ¹⁴+ζ¹⁷+ζ²³+ζ²⁶+ζ³²+ζ³⁵+ζ⁴¹+ζ⁴⁴+ζ⁵⁰+ζ⁵³+ζ⁵⁹+ζ⁶²+ζ⁶⁸+ζ⁷¹+ζ⁷⁷+ζ⁸⁰+ζ⁸⁶+ζ⁸⁹+ζ⁹⁸+ζ¹⁰⁴+ζ¹⁰⁷+ζ¹¹³+ζ¹¹⁶+ζ¹²²+ζ¹²⁵+ζ¹³¹+ζ¹³⁴+ζ¹⁴⁰+ζ¹⁴³+ζ¹⁴⁹+ζ¹⁵⁸+ζ¹⁶¹+ζ¹⁶⁷+ζ¹⁷⁰	−ζ⁸−ζ¹⁷−ζ²⁶−ζ³⁵−ζ⁴⁴−ζ⁵³−ζ⁶²−ζ⁷¹−ζ⁸⁰−ζ⁸⁹−ζ⁹⁸−ζ¹⁰⁷−ζ¹¹⁶−ζ¹²⁵−ζ¹³⁴−ζ¹⁴³−ζ¹⁶¹−ζ¹⁷⁰	ζ⁴+ζ⁷+ζ¹³+ζ¹⁶+ζ²²+ζ²⁵+ζ³¹+ζ³⁴+ζ⁴⁰+ζ⁴³+ζ⁴⁹+ζ⁵²+ζ⁵⁸+ζ⁶¹+ζ⁶⁷+ζ⁷⁰+ζ⁷⁹+ζ⁸⁵+ζ⁸⁸+ζ⁹⁴+ζ⁹⁷+ζ¹⁰³+ζ¹⁰⁶+ζ¹¹²+ζ¹¹⁵+ζ¹²¹+ζ¹²⁴+ζ¹³⁰+ζ¹³⁹+ζ¹⁴²+ζ¹⁴⁸+ζ¹⁵¹+ζ¹⁵⁷+ζ¹⁶⁰+ζ¹⁶⁶+ζ¹⁶⁹	−ζ⁷−ζ¹⁶−ζ²⁵−ζ³⁴−ζ⁴³−ζ⁵²−ζ⁶¹−ζ⁷⁰−ζ⁷⁹−ζ⁸⁸−ζ⁹⁷−ζ¹⁰⁶−ζ¹¹⁵−ζ¹²⁴−ζ¹⁴²−ζ¹⁵¹−ζ¹⁶⁰−ζ¹⁶⁹	−ζ⁴−ζ¹³−ζ²²−ζ³¹−ζ⁴⁰−ζ⁴⁹−ζ⁵⁸−ζ⁶⁷−ζ⁸⁵−ζ⁹⁴−ζ¹⁰³−ζ¹¹²−ζ¹²¹−ζ¹³⁰−ζ¹³⁹−ζ¹⁴⁸−ζ¹⁵⁷−ζ¹⁶⁶
χ₉	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	−ζ⁸−ζ¹⁷−ζ²⁶−ζ³⁵−ζ⁴⁴−ζ⁵³−ζ⁶²−ζ⁷¹−ζ⁸⁰−ζ⁸⁹−ζ⁹⁸−ζ¹⁰⁷−ζ¹¹⁶−ζ¹²⁵−ζ¹³⁴−ζ¹⁴³−ζ¹⁶¹−ζ¹⁷⁰	−ζ⁵−ζ¹⁴−ζ²³−ζ³²−ζ⁴¹−ζ⁵⁰−ζ⁵⁹−ζ⁶⁸−ζ⁷⁷−ζ⁸⁶−ζ¹⁰⁴−ζ¹¹³−ζ¹²²−ζ¹³¹−ζ¹⁴⁰−ζ¹⁴⁹−ζ¹⁵⁸−ζ¹⁶⁷	ζ⁵+ζ⁸+ζ¹⁴+ζ¹⁷+ζ²³+ζ²⁶+ζ³²+ζ³⁵+ζ⁴¹+ζ⁴⁴+ζ⁵⁰+ζ⁵³+ζ⁵⁹+ζ⁶²+ζ⁶⁸+ζ⁷¹+ζ⁷⁷+ζ⁸⁰+ζ⁸⁶+ζ⁸⁹+ζ⁹⁸+ζ¹⁰⁴+ζ¹⁰⁷+ζ¹¹³+ζ¹¹⁶+ζ¹²²+ζ¹²⁵+ζ¹³¹+ζ¹³⁴+ζ¹⁴⁰+ζ¹⁴³+ζ¹⁴⁹+ζ¹⁵⁸+ζ¹⁶¹+ζ¹⁶⁷+ζ¹⁷⁰	−ζ⁷−ζ¹⁶−ζ²⁵−ζ³⁴−ζ⁴³−ζ⁵²−ζ⁶¹−ζ⁷⁰−ζ⁷⁹−ζ⁸⁸−ζ⁹⁷−ζ¹⁰⁶−ζ¹¹⁵−ζ¹²⁴−ζ¹⁴²−ζ¹⁵¹−ζ¹⁶⁰−ζ¹⁶⁹	−ζ⁴−ζ¹³−ζ²²−ζ³¹−ζ⁴⁰−ζ⁴⁹−ζ⁵⁸−ζ⁶⁷−ζ⁸⁵−ζ⁹⁴−ζ¹⁰³−ζ¹¹²−ζ¹²¹−ζ¹³⁰−ζ¹³⁹−ζ¹⁴⁸−ζ¹⁵⁷−ζ¹⁶⁶	ζ⁴+ζ⁷+ζ¹³+ζ¹⁶+ζ²²+ζ²⁵+ζ³¹+ζ³⁴+ζ⁴⁰+ζ⁴³+ζ⁴⁹+ζ⁵²+ζ⁵⁸+ζ⁶¹+ζ⁶⁷+ζ⁷⁰+ζ⁷⁹+ζ⁸⁵+ζ⁸⁸+ζ⁹⁴+ζ⁹⁷+ζ¹⁰³+ζ¹⁰⁶+ζ¹¹²+ζ¹¹⁵+ζ¹²¹+ζ¹²⁴+ζ¹³⁰+ζ¹³⁹+ζ¹⁴²+ζ¹⁴⁸+ζ¹⁵¹+ζ¹⁵⁷+ζ¹⁶⁰+ζ¹⁶⁶+ζ¹⁶⁹
χ₁₀	9	$\frac{- 1 + i \sqrt{19}}{2}$	$\frac{- 1 - i \sqrt{19}}{2}$	0	0	0	0	0	0	0	0
χ₁₁	9	$\frac{- 1 - i \sqrt{19}}{2}$	$\frac{- 1 + i \sqrt{19}}{2}$	0	0	0	0	0	0	0	0

Here ζ is a primitive root of unity of order 171 (ζ¹⁷¹ = 1); i = ζ⁴².

Relationships

maximal subgroup of	AGL(1,𝔽₁₉), C₂×#4296, C₃×#4296, …
maximal normal subgroup of	AGL(1,𝔽₁₉), C₂×#4296, C₃×#4296, …
maximal quotient of	C₂×#4296, #10587020, C₃×#4296, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(171, 3);