#30681047

order	16807
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₇² · Schur multiplier (H₂) = C₇

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₇²	H(3,𝔽₇)
commutator, Frattini, cosocle	C₇³	C₇²
prime radical	C₇×H(3,𝔽₇)	C₇
LCS	G ▷ C₇³ ▷ C₇² ▷ C₁
UCS	C₁ ◁ C₇² ◁ C₇³ ◁ G
LECS	G ▷ C₇³ ▷ C₇² ▷ C₁

85 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 98
transitive degree	μ_t(G) = 2401

A minimal faithful permutation representation — 2 generators on 98 points:

(1,2,4,8,16,27,35)(3,5,11,19,29,36,44)(6,13,17,30,40,47,43)(7,12,23,32,39,37,45)(9,20,28,38,46,15,24)(10,22,33,42,49,34,14)(18,31,41,48,26,25,21)(51,53,57,65,73,63,55)(54,59,62,61,69,58,66)(60,70,78,75,68,72,67)(71,77,79,82,85,76,81)(80,84,91,86,90,89,87);(2,5,12,24,25,14,6)(4,9,13,23,10,11,21)(8,17,22,18,20,32,19)(16,28,30,39,33,29,31)(27,36,37,38,41,42,40)(50,51,54,60,71,80,88,52,55,62,70,79,87,95,56,63,69,78,85,89,96,64,73,66,75,81,90,97,74,65,59,68,77,86,94,83,57,61,72,82,91,98,92,53,58,67,76,84,93)

Polycyclic presentation:

7^5|b^7=e,b^a=bc,c^a=cd,c^b=ce

Conjugacy classes & character table

391 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16807, 4);