C₅×H(3,𝔽₇) #13133827

order	1715
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₇)
automorphism	?
homology	Abelianization (H₁) = C₅×C₇² · Schur multiplier (H₂) = C₇²

Aliases

direct product	C₅×H(3,𝔽₇)
unambiguous semidirect product	C₇²⋊(C₅×C₇)
unambiguous semidirect product	(C₅×C₇²)⋊C₇

Subgroups

	subgroup	G/H
center, socle	C₅×C₇	C₇²
commutator, Frattini, cosocle	C₇	C₅×C₇²
LCS	G ▷ C₇ ▷ C₁
UCS	C₁ ◁ C₅×C₇ ◁ G

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 245

A minimal faithful permutation representation — 2 generators on 54 points:

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

Conjugacy classes & character table

275 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×H(3,𝔽₇)⋊C₇, C₅×#14558878, C₅×#14558879, …
maximal normal subgroup of	C₅×H(3,𝔽₇)⋊C₇, C₅×#14558878, C₅×#14558879, …
maximal quotient of	C₅×C₇²⋊C₄₉, C₅×H(3,𝔽₇)⋊C₇, C₅×#14558877, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1715, 3);

C5×H(3,𝔽7) #13133827