SL(2,𝔽₅)×SL(2,𝔽₇) #72227514

order	40320
isoclinism	isoclinic class of SL(2,𝔽₅)×SL(2,𝔽₇)
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₁

Aliases

direct product

SL(2,𝔽₅)×SL(2,𝔽₇)

Subgroups

	subgroup	G/H
center, Frattini, Fitting, socle, cosocle, hypercenter	C₂²	A₅×GL(3,𝔽₂)
commutator, hypocenter	G	C₁
LCS	G
UCS	C₁ ◁ C₂²

374 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 2 generators on 40 points:

(1,5,4,10)(2,13,7,9)(3,6,8,11)(12,15,16,14)(18,23,28,30,25)(19,24,29,21,22)(26,32,33,37,36)(27,34,38,35,31);(1,12,14,4,16,15)(2,7)(3,9,5,8,13,10)(6,11)(17,29,36,31,18)(19,28,37,40,27)(20,30,35,26,22)(23,24,38,39,32)

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

GAP

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

gap> G := Group((1,5,4,10)(2,13,7,9)(3,6,8,11)(12,15,16,14)(18,23,28,30,25)(19,24,29,21,22)(26,32,33,37,36)(27,34,38,35,31), (1,12,14,4,16,15)(2,7)(3,9,5,8,13,10)(6,11)(17,29,36,31,18)(19,28,37,40,27)(20,30,35,26,22)(23,24,38,39,32));