#401⋊C₇ #89417

order	448
isoclinism	isoclinic class of #401⋊C₇
automorphism	Aut(G) = (#401⋊C₇)⋊C₃ (order 1344) · Inn(G) = G · Out(G) = C₃
homology	Abelianization (H₁) = C₇ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product

#401⋊C₇

Splittings

Fitting, derived, Sylow

#401 ⋊ C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	#401	C₇
Frattini, socle	C₂³	AGL(1,𝔽₈)
LCS	G ▷ #401
UCS	C₁

14 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 56 points:

(1,2,6,14,24,32,41)(3,8,17,29,39,47,55)(4,9,22,31,40,48,49)(5,10,18,27,35,44,51)(7,16,26,37,46,54,13)(11,20,15,25,34,43,52)(12,21,30,38,33,42,50)(19,28,36,45,53,56,23);(1,23,5,11)(2,20,10,19)(3,13,12,4)(6,17,18,30)(7,21,9,8)(15,22,28,16)(24,40,35,37)(26,31)(29,38)(32,42)(33,45,39,34)(41,49)(43,46)(44,47)(48,53)(50,52)(51,54)(55,56)

Polycyclic presentation:

7,2^6|b^2=efg,c^2=ef,d^2=e,b^a=c,c^a=d,d^a=bd,e^a=g,f^a=eg,g^a=f,c^b=ce,d^b=df,d^c=dg

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	7	28	28	64	64	64	64	64	64
order	1	2	4	4	7	7	7	7	7	7
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	−ζ²	−ζ¹⁸	−ζ⁶	−ζ²²	−ζ¹⁰	−ζ²⁶
χ₃	1	1	1	1	−ζ¹⁸	−ζ²²	−ζ²⁶	−ζ²	−ζ⁶	−ζ¹⁰
χ₄	1	1	1	1	−ζ⁶	−ζ²⁶	−ζ¹⁸	−ζ¹⁰	−ζ²	−ζ²²
χ₅	1	1	1	1	−ζ²²	−ζ²	−ζ¹⁰	−ζ¹⁸	−ζ²⁶	−ζ⁶
χ₆	1	1	1	1	−ζ¹⁰	−ζ⁶	−ζ²	−ζ²⁶	−ζ²²	−ζ¹⁸
χ₇	1	1	1	1	−ζ²⁶	−ζ¹⁰	−ζ²²	−ζ⁶	−ζ¹⁸	−ζ²
χ₈	7	7	-1	-1	0	0	0	0	0	0
χ₉	14	-2	$2 i$	$- 2 i$	0	0	0	0	0	0
χ₁₀	14	-2	$- 2 i$	$2 i$	0	0	0	0	0	0

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

Relationships

derived subgroup of	(#401⋊C₇)⋊C₃
maximal subgroup of	C₂×#401⋊C₇, (#401⋊C₇)⋊C₃, C₃×#401⋊C₇, …
maximal normal subgroup of	C₂×#401⋊C₇, (#401⋊C₇)⋊C₃, C₃×#401⋊C₇
maximal quotient of	#2104⋊C₇, C₂×#401⋊C₇, C₃×#401⋊C₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 179);

#401⋊C7 #89417