C₇×Sp(2,𝔽₁₃) #72234263

order	15288
isoclinism	isoclinic class of Sp(2,𝔽₁₃)
automorphism	?
homology	Abelianization (H₁) = C₇ · Schur multiplier (H₂) = C₁

non-solvable

Aliases

direct product

C₇×Sp(2,𝔽₁₃)

Subgroups

	subgroup	G/H
center, Fitting, socle, hypercenter	C₂×C₇	PSL(2,𝔽₁₃)
commutator, hypocenter	Sp(2,𝔽₁₃)	C₇
Frattini, cosocle	C₂	C₇×PSL(2,𝔽₁₃)
LCS	G ▷ Sp(2,𝔽₁₃)
UCS	C₁ ◁ C₂×C₇

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 392

A minimal faithful permutation representation — 2 generators on 63 points:

(1,6,4,2,7,5,3)(8,33,60)(10,16,56)(11,49,22)(12,61,14)(13,29,24)(15,47,36)(17,31,45)(18,55,28)(19,46,53)(21,58,39)(25,30,42)(26,54,57)(27,59,44)(34,51,63)(35,38,62)(37,41,48);(1,5,2,6,3,7,4)(8,31,30,56,27,52,48,29,55,54,11,14,50,58)(9,60,37,47,46,12,16,43,13,39,35,34,59,22)(10,51,26,32,18,38,33,49,53,25,23,17,36,24)(15,28,21,61,42,63,40,62,45,41,44,57,19,20)

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₇×#72234266
maximal normal subgroup of	C₇×#72234266

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,6,4,2,7,5,3)(8,33,60)(10,16,56)(11,49,22)(12,61,14)(13,29,24)(15,47,36)(17,31,45)(18,55,28)(19,46,53)(21,58,39)(25,30,42)(26,54,57)(27,59,44)(34,51,63)(35,38,62)(37,41,48), (1,5,2,6,3,7,4)(8,31,30,56,27,52,48,29,55,54,11,14,50,58)(9,60,37,47,46,12,16,43,13,39,35,34,59,22)(10,51,26,32,18,38,33,49,53,25,23,17,36,24)(15,28,21,61,42,63,40,62,45,41,44,57,19,20));

C7×Sp(2,𝔽13) #72234263