#6729⋊C₈ #14514982

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

non-solvable

Aliases

unambiguous semidirect product

#6729⋊C₈

Splittings

derived

SL(2,𝔽₅) ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×S₅
commutator, hypocenter	SL(2,𝔽₅)	C₂×C₈
Fitting, hypercenter	C₂×C₈	S₅
socle	C₂²	C₄×S₅
prime radical	C₂×#6729	C₄
cosocle	C₄×SL(2,𝔽₅)	C₂²
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

198 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 320

Permutation generators:

(1,40,2,39)(3,37,4,38)(5,34,6,33)(7,36,8,35)(9,48,19,55,10,47,20,56)(11,45,18,54,12,46,17,53)(13,52,22,41,14,51,21,42)(15,49,24,43,16,50,23,44)(25,58,28,59,26,57,27,60)(29,64,31,61,30,63,32,62);(1,33,3,35,2,34,4,36)(5,37,7,39,6,38,8,40)(9,53,17,64,26,41,23,49,29,58,11,55,19,61,28,43,22,51,31,60,10,54,18,63,25,42,24,50,30,57,12,56,20,62,27,44,21,52,32,59)(13,46,15,48,14,45,16,47)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 240680);