C₇×H(3,𝔽₇) #14558881

order	2401
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₇)
automorphism	?
homology	Abelianization (H₁) = C₇³ · Schur multiplier (H₂) = C₇⁴

Aliases

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

Subgroups

	subgroup	G/H
center, socle	C₇²	C₇²
commutator, Frattini, cosocle	C₇	C₇³
LCS	G ▷ C₇ ▷ C₁
UCS	C₁ ◁ C₇² ◁ G
LECS	G ▷ C₇ ▷ C₁

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 343

A minimal faithful permutation representation — 3 generators on 56 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₇×H(3,𝔽₇), C₇×(C₇⋊D₇)⋊C₇, D₇×H(3,𝔽₇), …
maximal normal subgroup of	C₂×C₇×H(3,𝔽₇), C₇×(C₇⋊D₇)⋊C₇, D₇×H(3,𝔽₇), …
maximal quotient of	C₂×C₇×H(3,𝔽₇), C₅×C₇×H(3,𝔽₇), C₇²⋊H(3,𝔽₇), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2401, 12);

C7×H(3,𝔽7) #14558881