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#3633

order	136
isoclinism	isoclinic class of #3633
automorphism	Aut(G) = AGL(1,𝔽₁₇) (order 272) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72243029
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow

C₁₇ ⋊ C₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₁₇	C₈
prime radical	D₁₇	C₄
cosocle	#595	C₂
LCS	G ▷ C₁₇
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 17

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

2^3,17|a^2=b,b^2=c,d^a=d^9,d^b=d^13,d^c=d^16

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	8	8	17	17	17	17	17	17	17
order	1	17	17	2	4	4	8	8	8	8
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	-1	-1	-1	-1
χ₃	1	1	1	1	-1	-1	$i$	$i$	$- i$	$- i$
χ₄	1	1	1	1	-1	-1	$- i$	$- i$	$i$	$i$
χ₅	1	1	1	-1	$i$	$- i$	ζ⁵+ζ¹³+ζ²¹+ζ²⁹+ζ³⁷+ζ⁴⁵+ζ⁵³+ζ⁶¹+ζ⁶⁹+ζ⁷⁷+ζ⁹³+ζ¹⁰¹+ζ¹⁰⁹+ζ¹¹⁷+ζ¹²⁵+ζ¹³³	−ζ⁵−ζ¹³−ζ²¹−ζ²⁹−ζ³⁷−ζ⁴⁵−ζ⁵³−ζ⁶¹−ζ⁶⁹−ζ⁷⁷−ζ⁹³−ζ¹⁰¹−ζ¹⁰⁹−ζ¹¹⁷−ζ¹²⁵−ζ¹³³	ζ⁷+ζ¹⁵+ζ²³+ζ³¹+ζ³⁹+ζ⁴⁷+ζ⁵⁵+ζ⁶³+ζ⁷¹+ζ⁷⁹+ζ⁸⁷+ζ⁹⁵+ζ¹⁰³+ζ¹¹¹+ζ¹²⁷+ζ¹³⁵	−ζ⁷−ζ¹⁵−ζ²³−ζ³¹−ζ³⁹−ζ⁴⁷−ζ⁵⁵−ζ⁶³−ζ⁷¹−ζ⁷⁹−ζ⁸⁷−ζ⁹⁵−ζ¹⁰³−ζ¹¹¹−ζ¹²⁷−ζ¹³⁵
χ₆	1	1	1	-1	$i$	$- i$	−ζ⁵−ζ¹³−ζ²¹−ζ²⁹−ζ³⁷−ζ⁴⁵−ζ⁵³−ζ⁶¹−ζ⁶⁹−ζ⁷⁷−ζ⁹³−ζ¹⁰¹−ζ¹⁰⁹−ζ¹¹⁷−ζ¹²⁵−ζ¹³³	ζ⁵+ζ¹³+ζ²¹+ζ²⁹+ζ³⁷+ζ⁴⁵+ζ⁵³+ζ⁶¹+ζ⁶⁹+ζ⁷⁷+ζ⁹³+ζ¹⁰¹+ζ¹⁰⁹+ζ¹¹⁷+ζ¹²⁵+ζ¹³³	−ζ⁷−ζ¹⁵−ζ²³−ζ³¹−ζ³⁹−ζ⁴⁷−ζ⁵⁵−ζ⁶³−ζ⁷¹−ζ⁷⁹−ζ⁸⁷−ζ⁹⁵−ζ¹⁰³−ζ¹¹¹−ζ¹²⁷−ζ¹³⁵	ζ⁷+ζ¹⁵+ζ²³+ζ³¹+ζ³⁹+ζ⁴⁷+ζ⁵⁵+ζ⁶³+ζ⁷¹+ζ⁷⁹+ζ⁸⁷+ζ⁹⁵+ζ¹⁰³+ζ¹¹¹+ζ¹²⁷+ζ¹³⁵
χ₇	1	1	1	-1	$- i$	$i$	ζ⁷+ζ¹⁵+ζ²³+ζ³¹+ζ³⁹+ζ⁴⁷+ζ⁵⁵+ζ⁶³+ζ⁷¹+ζ⁷⁹+ζ⁸⁷+ζ⁹⁵+ζ¹⁰³+ζ¹¹¹+ζ¹²⁷+ζ¹³⁵	−ζ⁷−ζ¹⁵−ζ²³−ζ³¹−ζ³⁹−ζ⁴⁷−ζ⁵⁵−ζ⁶³−ζ⁷¹−ζ⁷⁹−ζ⁸⁷−ζ⁹⁵−ζ¹⁰³−ζ¹¹¹−ζ¹²⁷−ζ¹³⁵	ζ⁵+ζ¹³+ζ²¹+ζ²⁹+ζ³⁷+ζ⁴⁵+ζ⁵³+ζ⁶¹+ζ⁶⁹+ζ⁷⁷+ζ⁹³+ζ¹⁰¹+ζ¹⁰⁹+ζ¹¹⁷+ζ¹²⁵+ζ¹³³	−ζ⁵−ζ¹³−ζ²¹−ζ²⁹−ζ³⁷−ζ⁴⁵−ζ⁵³−ζ⁶¹−ζ⁶⁹−ζ⁷⁷−ζ⁹³−ζ¹⁰¹−ζ¹⁰⁹−ζ¹¹⁷−ζ¹²⁵−ζ¹³³
χ₈	1	1	1	-1	$- i$	$i$	−ζ⁷−ζ¹⁵−ζ²³−ζ³¹−ζ³⁹−ζ⁴⁷−ζ⁵⁵−ζ⁶³−ζ⁷¹−ζ⁷⁹−ζ⁸⁷−ζ⁹⁵−ζ¹⁰³−ζ¹¹¹−ζ¹²⁷−ζ¹³⁵	ζ⁷+ζ¹⁵+ζ²³+ζ³¹+ζ³⁹+ζ⁴⁷+ζ⁵⁵+ζ⁶³+ζ⁷¹+ζ⁷⁹+ζ⁸⁷+ζ⁹⁵+ζ¹⁰³+ζ¹¹¹+ζ¹²⁷+ζ¹³⁵	−ζ⁵−ζ¹³−ζ²¹−ζ²⁹−ζ³⁷−ζ⁴⁵−ζ⁵³−ζ⁶¹−ζ⁶⁹−ζ⁷⁷−ζ⁹³−ζ¹⁰¹−ζ¹⁰⁹−ζ¹¹⁷−ζ¹²⁵−ζ¹³³	ζ⁵+ζ¹³+ζ²¹+ζ²⁹+ζ³⁷+ζ⁴⁵+ζ⁵³+ζ⁶¹+ζ⁶⁹+ζ⁷⁷+ζ⁹³+ζ¹⁰¹+ζ¹⁰⁹+ζ¹¹⁷+ζ¹²⁵+ζ¹³³
χ₉	8	$\frac{- 1 - \sqrt{17}}{2}$	$\frac{- 1 + \sqrt{17}}{2}$	0	0	0	0	0	0	0
χ₁₀	8	$\frac{- 1 + \sqrt{17}}{2}$	$\frac{- 1 - \sqrt{17}}{2}$	0	0	0	0	0	0	0

Here ζ is a primitive root of unity of order 136 (ζ¹³⁶ = 1); i = ζ³⁴.

Relationships

maximal subgroup of	AGL(1,𝔽₁₇), C₂×#3633, C₃×#3633, …
maximal normal subgroup of	AGL(1,𝔽₁₇), C₂×#3633, C₃×#3633, …
maximal quotient of	#63241, C₂×#3633, C₃×#3633, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(136, 12);