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D₃₇⋊C₃ #6362

order	222
isoclinism	isoclinic class of D₃₇⋊C₃
automorphism	Aut(G) = AGL(1,𝔽₃₇) (order 1332) · Inn(G) = G · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	D₃₇⋊C₃
unambiguous semidirect product	(C₃₇⋊C₃)⋊C₂

Splittings

Fitting, socle, derived, Sylow	C₃₇ ⋊ C₂×C₃
Hall	C₃₇⋊C₃ ⋊ C₂
Hall	D₃₇ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃₇	C₂×C₃
LCS	G ▷ C₃₇
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 37

A minimal faithful permutation representation — 2 generators on 37 points:

(2,6,7,3,4,8)(5,14,15,9,10,23)(11,26,27,17,18,34)(12,25,21,19,16,13)(20,36,37,29,28,32)(22,24,35,31,30,33);(1,2,5,11,18,28,33,34,31,21,7,6,12,20,30,10,16,23,32,37,15,25,14,24,29,19,4,8,13,22,27,35,36,26,17,9,3)

Polycyclic presentation:

2,3,37|c^a=c^36,c^b=c^26

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	6	6	6	6	6	6	37	37	37	37	37
order	1	37	37	37	37	37	37	2	3	3	6	6
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	1	1	1	1	1	1	1	-1	1	1	-1	-1
χ₅	1	1	1	1	1	1	1	-1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₆	1	1	1	1	1	1	1	-1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₇	6	−ζ⁸−ζ¹⁴−ζ²³−ζ²⁹−ζ³¹−ζ⁴³−ζ⁶⁸−ζ⁸⁰−ζ⁸²−ζ⁸⁸−ζ⁹⁷−ζ¹⁰³	−ζ−ζ¹⁰−ζ¹¹−ζ²⁶−ζ³⁸−ζ⁴⁷−ζ⁶⁴−ζ⁷³−ζ⁸⁵−ζ¹⁰⁰−ζ¹⁰¹−ζ¹¹⁰	−ζ⁵−ζ¹³−ζ¹⁹−ζ³²−ζ⁵⁰−ζ⁵⁵−ζ⁵⁶−ζ⁶¹−ζ⁷⁹−ζ⁹²−ζ⁹⁸−ζ¹⁰⁶	−ζ⁴−ζ⁷−ζ³⁴−ζ⁴⁰−ζ⁴¹−ζ⁴⁴−ζ⁶⁷−ζ⁷⁰−ζ⁷¹−ζ⁷⁷−ζ¹⁰⁴−ζ¹⁰⁷	−ζ¹⁶−ζ²⁵−ζ²⁸−ζ⁴⁶−ζ⁴⁹−ζ⁵³−ζ⁵⁸−ζ⁶²−ζ⁶⁵−ζ⁸³−ζ⁸⁶−ζ⁹⁵	−ζ²−ζ¹⁷−ζ²⁰−ζ²²−ζ³⁵−ζ⁵²−ζ⁵⁹−ζ⁷⁶−ζ⁸⁹−ζ⁹¹−ζ⁹⁴−ζ¹⁰⁹	0	0	0	0	0
χ₈	6	−ζ−ζ¹⁰−ζ¹¹−ζ²⁶−ζ³⁸−ζ⁴⁷−ζ⁶⁴−ζ⁷³−ζ⁸⁵−ζ¹⁰⁰−ζ¹⁰¹−ζ¹¹⁰	−ζ⁸−ζ¹⁴−ζ²³−ζ²⁹−ζ³¹−ζ⁴³−ζ⁶⁸−ζ⁸⁰−ζ⁸²−ζ⁸⁸−ζ⁹⁷−ζ¹⁰³	−ζ⁴−ζ⁷−ζ³⁴−ζ⁴⁰−ζ⁴¹−ζ⁴⁴−ζ⁶⁷−ζ⁷⁰−ζ⁷¹−ζ⁷⁷−ζ¹⁰⁴−ζ¹⁰⁷	−ζ⁵−ζ¹³−ζ¹⁹−ζ³²−ζ⁵⁰−ζ⁵⁵−ζ⁵⁶−ζ⁶¹−ζ⁷⁹−ζ⁹²−ζ⁹⁸−ζ¹⁰⁶	−ζ²−ζ¹⁷−ζ²⁰−ζ²²−ζ³⁵−ζ⁵²−ζ⁵⁹−ζ⁷⁶−ζ⁸⁹−ζ⁹¹−ζ⁹⁴−ζ¹⁰⁹	−ζ¹⁶−ζ²⁵−ζ²⁸−ζ⁴⁶−ζ⁴⁹−ζ⁵³−ζ⁵⁸−ζ⁶²−ζ⁶⁵−ζ⁸³−ζ⁸⁶−ζ⁹⁵	0	0	0	0	0
χ₉	6	−ζ⁵−ζ¹³−ζ¹⁹−ζ³²−ζ⁵⁰−ζ⁵⁵−ζ⁵⁶−ζ⁶¹−ζ⁷⁹−ζ⁹²−ζ⁹⁸−ζ¹⁰⁶	−ζ⁴−ζ⁷−ζ³⁴−ζ⁴⁰−ζ⁴¹−ζ⁴⁴−ζ⁶⁷−ζ⁷⁰−ζ⁷¹−ζ⁷⁷−ζ¹⁰⁴−ζ¹⁰⁷	−ζ²−ζ¹⁷−ζ²⁰−ζ²²−ζ³⁵−ζ⁵²−ζ⁵⁹−ζ⁷⁶−ζ⁸⁹−ζ⁹¹−ζ⁹⁴−ζ¹⁰⁹	−ζ¹⁶−ζ²⁵−ζ²⁸−ζ⁴⁶−ζ⁴⁹−ζ⁵³−ζ⁵⁸−ζ⁶²−ζ⁶⁵−ζ⁸³−ζ⁸⁶−ζ⁹⁵	−ζ−ζ¹⁰−ζ¹¹−ζ²⁶−ζ³⁸−ζ⁴⁷−ζ⁶⁴−ζ⁷³−ζ⁸⁵−ζ¹⁰⁰−ζ¹⁰¹−ζ¹¹⁰	−ζ⁸−ζ¹⁴−ζ²³−ζ²⁹−ζ³¹−ζ⁴³−ζ⁶⁸−ζ⁸⁰−ζ⁸²−ζ⁸⁸−ζ⁹⁷−ζ¹⁰³	0	0	0	0	0
χ₁₀	6	−ζ⁴−ζ⁷−ζ³⁴−ζ⁴⁰−ζ⁴¹−ζ⁴⁴−ζ⁶⁷−ζ⁷⁰−ζ⁷¹−ζ⁷⁷−ζ¹⁰⁴−ζ¹⁰⁷	−ζ⁵−ζ¹³−ζ¹⁹−ζ³²−ζ⁵⁰−ζ⁵⁵−ζ⁵⁶−ζ⁶¹−ζ⁷⁹−ζ⁹²−ζ⁹⁸−ζ¹⁰⁶	−ζ¹⁶−ζ²⁵−ζ²⁸−ζ⁴⁶−ζ⁴⁹−ζ⁵³−ζ⁵⁸−ζ⁶²−ζ⁶⁵−ζ⁸³−ζ⁸⁶−ζ⁹⁵	−ζ²−ζ¹⁷−ζ²⁰−ζ²²−ζ³⁵−ζ⁵²−ζ⁵⁹−ζ⁷⁶−ζ⁸⁹−ζ⁹¹−ζ⁹⁴−ζ¹⁰⁹	−ζ⁸−ζ¹⁴−ζ²³−ζ²⁹−ζ³¹−ζ⁴³−ζ⁶⁸−ζ⁸⁰−ζ⁸²−ζ⁸⁸−ζ⁹⁷−ζ¹⁰³	−ζ−ζ¹⁰−ζ¹¹−ζ²⁶−ζ³⁸−ζ⁴⁷−ζ⁶⁴−ζ⁷³−ζ⁸⁵−ζ¹⁰⁰−ζ¹⁰¹−ζ¹¹⁰	0	0	0	0	0
χ₁₁	6	−ζ¹⁶−ζ²⁵−ζ²⁸−ζ⁴⁶−ζ⁴⁹−ζ⁵³−ζ⁵⁸−ζ⁶²−ζ⁶⁵−ζ⁸³−ζ⁸⁶−ζ⁹⁵	−ζ²−ζ¹⁷−ζ²⁰−ζ²²−ζ³⁵−ζ⁵²−ζ⁵⁹−ζ⁷⁶−ζ⁸⁹−ζ⁹¹−ζ⁹⁴−ζ¹⁰⁹	−ζ−ζ¹⁰−ζ¹¹−ζ²⁶−ζ³⁸−ζ⁴⁷−ζ⁶⁴−ζ⁷³−ζ⁸⁵−ζ¹⁰⁰−ζ¹⁰¹−ζ¹¹⁰	−ζ⁸−ζ¹⁴−ζ²³−ζ²⁹−ζ³¹−ζ⁴³−ζ⁶⁸−ζ⁸⁰−ζ⁸²−ζ⁸⁸−ζ⁹⁷−ζ¹⁰³	−ζ⁵−ζ¹³−ζ¹⁹−ζ³²−ζ⁵⁰−ζ⁵⁵−ζ⁵⁶−ζ⁶¹−ζ⁷⁹−ζ⁹²−ζ⁹⁸−ζ¹⁰⁶	−ζ⁴−ζ⁷−ζ³⁴−ζ⁴⁰−ζ⁴¹−ζ⁴⁴−ζ⁶⁷−ζ⁷⁰−ζ⁷¹−ζ⁷⁷−ζ¹⁰⁴−ζ¹⁰⁷	0	0	0	0	0
χ₁₂	6	−ζ²−ζ¹⁷−ζ²⁰−ζ²²−ζ³⁵−ζ⁵²−ζ⁵⁹−ζ⁷⁶−ζ⁸⁹−ζ⁹¹−ζ⁹⁴−ζ¹⁰⁹	−ζ¹⁶−ζ²⁵−ζ²⁸−ζ⁴⁶−ζ⁴⁹−ζ⁵³−ζ⁵⁸−ζ⁶²−ζ⁶⁵−ζ⁸³−ζ⁸⁶−ζ⁹⁵	−ζ⁸−ζ¹⁴−ζ²³−ζ²⁹−ζ³¹−ζ⁴³−ζ⁶⁸−ζ⁸⁰−ζ⁸²−ζ⁸⁸−ζ⁹⁷−ζ¹⁰³	−ζ−ζ¹⁰−ζ¹¹−ζ²⁶−ζ³⁸−ζ⁴⁷−ζ⁶⁴−ζ⁷³−ζ⁸⁵−ζ¹⁰⁰−ζ¹⁰¹−ζ¹¹⁰	−ζ⁴−ζ⁷−ζ³⁴−ζ⁴⁰−ζ⁴¹−ζ⁴⁴−ζ⁶⁷−ζ⁷⁰−ζ⁷¹−ζ⁷⁷−ζ¹⁰⁴−ζ¹⁰⁷	−ζ⁵−ζ¹³−ζ¹⁹−ζ³²−ζ⁵⁰−ζ⁵⁵−ζ⁵⁶−ζ⁶¹−ζ⁷⁹−ζ⁹²−ζ⁹⁸−ζ¹⁰⁶	0	0	0	0	0

Here ζ is a primitive root of unity of order 111 (ζ¹¹¹ = 1); i = ζ²⁷.

Relationships

maximal subgroup of	#3866⋊C₃, C₂×D₃₇⋊C₃, #66526⋊C₂, …
maximal normal subgroup of	#3866⋊C₃, C₂×D₃₇⋊C₃, #66526⋊C₂, …
maximal quotient of	Dic₃₇⋊C₃, C₂×D₃₇⋊C₃, #66524⋊C₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(222, 1);