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D₂₁ #187

order	42
isoclinism	isoclinic class of D₂₁
automorphism	Aut(G) = D₃×AGL(1,𝔽₇) (order 252) · Inn(G) = G · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = #72213636
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₂₁
unambiguous semidirect product	C₃⋊D₇
unambiguous semidirect product	C₇⋊D₃

Splittings

Fitting, socle, derived, Hall	C₃×C₇ ⋊ C₂
Sylow	C₇ ⋊ D₃
Sylow	C₃ ⋊ D₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇	C₂
LCS	G ▷ C₃×C₇
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 21

A minimal faithful permutation representation — 2 generators on 10 points:

(1,2,3)(4,5,6,7,8,9,10);(2,3)(5,10)(6,9)(7,8)

Polycyclic presentation:

2,3,7|b^a=b^2,c^a=c^6

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	2	2	2	2	2	2	2	2	2	2	21
order	1	3	7	7	7	21	21	21	21	21	21	2
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	1	1	1	1	-1
χ₃	2	2	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	0
χ₄	2	2	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ−ζ⁸−ζ¹³−ζ²⁰	0
χ₅	2	2	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	0
χ₆	2	-1	2	2	2	-1	-1	-1	-1	-1	-1	0
χ₇	2	-1	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	ζ+ζ²⁰	ζ⁸+ζ¹³	ζ¹⁰+ζ¹¹	ζ⁴+ζ¹⁷	ζ⁵+ζ¹⁶	ζ²+ζ¹⁹	0
χ₈	2	-1	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	ζ⁸+ζ¹³	ζ+ζ²⁰	ζ⁴+ζ¹⁷	ζ¹⁰+ζ¹¹	ζ²+ζ¹⁹	ζ⁵+ζ¹⁶	0
χ₉	2	-1	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ−ζ⁸−ζ¹³−ζ²⁰	ζ¹⁰+ζ¹¹	ζ⁴+ζ¹⁷	ζ⁵+ζ¹⁶	ζ²+ζ¹⁹	ζ⁸+ζ¹³	ζ+ζ²⁰	0
χ₁₀	2	-1	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ−ζ⁸−ζ¹³−ζ²⁰	ζ⁴+ζ¹⁷	ζ¹⁰+ζ¹¹	ζ²+ζ¹⁹	ζ⁵+ζ¹⁶	ζ+ζ²⁰	ζ⁸+ζ¹³	0
χ₁₁	2	-1	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	ζ⁵+ζ¹⁶	ζ²+ζ¹⁹	ζ⁸+ζ¹³	ζ+ζ²⁰	ζ⁴+ζ¹⁷	ζ¹⁰+ζ¹¹	0
χ₁₂	2	-1	−ζ²−ζ⁵−ζ¹⁶−ζ¹⁹	−ζ−ζ⁸−ζ¹³−ζ²⁰	−ζ⁴−ζ¹⁰−ζ¹¹−ζ¹⁷	ζ²+ζ¹⁹	ζ⁵+ζ¹⁶	ζ+ζ²⁰	ζ⁸+ζ¹³	ζ¹⁰+ζ¹¹	ζ⁴+ζ¹⁷	0

Here ζ is a primitive root of unity of order 21 (ζ²¹ = 1).

Relationships

maximal subgroup of	D₃×D₇, C₂×D₂₁, D₆₃, …
maximal normal subgroup of	D₃×D₇, C₂×D₂₁, C₃⋊AGL(1,𝔽₇), …
maximal quotient of	Dic₂₁, C₂×D₂₁, D₆₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(42, 5);