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Dic₇ #84

order	28
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂×AGL(1,𝔽₇) (order 84) · Inn(G) = D₇ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = #72210376
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇
unambiguous semidirect product	C₇⋊C₄

Splittings

C₇ ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇
commutator, hypocenter	C₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 2 generators on 11 points:

(1,2)(3,4)(5,6,7,8,9,10,11);(1,3,2,4)(6,11)(7,10)(8,9)

Polycyclic presentation:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	1	2	2	2	2	2	2	7	7
order	1	2	7	7	7	14	14	14	4	4
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	-1	-1	$i$	$- i$
χ₄	1	-1	1	1	1	-1	-1	-1	$- i$	$i$
χ₅	2	2	−ζ¹⁰−ζ¹⁸	−ζ⁶−ζ²²	−ζ²−ζ²⁶	−ζ¹⁰−ζ¹⁸	−ζ⁶−ζ²²	−ζ²−ζ²⁶	0	0
χ₆	2	2	−ζ⁶−ζ²²	−ζ²−ζ²⁶	−ζ¹⁰−ζ¹⁸	−ζ⁶−ζ²²	−ζ²−ζ²⁶	−ζ¹⁰−ζ¹⁸	0	0
χ₇	2	2	−ζ²−ζ²⁶	−ζ¹⁰−ζ¹⁸	−ζ⁶−ζ²²	−ζ²−ζ²⁶	−ζ¹⁰−ζ¹⁸	−ζ⁶−ζ²²	0	0
χ₈	2	-2	−ζ¹⁰−ζ¹⁸	−ζ⁶−ζ²²	−ζ²−ζ²⁶	ζ¹⁰+ζ¹⁸	ζ⁶+ζ²²	ζ²+ζ²⁶	0	0
χ₉	2	-2	−ζ⁶−ζ²²	−ζ²−ζ²⁶	−ζ¹⁰−ζ¹⁸	ζ⁶+ζ²²	ζ²+ζ²⁶	ζ¹⁰+ζ¹⁸	0	0
χ₁₀	2	-2	−ζ²−ζ²⁶	−ζ¹⁰−ζ¹⁸	−ζ⁶−ζ²²	ζ²+ζ²⁶	ζ¹⁰+ζ¹⁸	ζ⁶+ζ²²	0	0

Here ζ is a primitive root of unity of order 28 (ζ²⁸ = 1); i = ζ⁷.

Relationships

maximal subgroup of	Dic₁₄, C₄×D₇, C₂×Dic₇, …
maximal normal subgroup of	Dic₁₄, C₄×D₇, C₂×Dic₇, …
maximal quotient of	C₇⋊C₈, C₂×Dic₇, C₃×Dic₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28, 1);