C₅×#85370 #14511108

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product

C₅×#85370

Splittings

Fitting

C₅×#520 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₅×#520	D₃
prime radical	C₂×C₅×#985	C₂
hypercenter	C₅×C₄⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

516 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(6,7,8,9)(10,23,28)(11,18,33,17,27,32)(12,21,39,20,34,37)(13,14,26)(16,38,41)(19,25)(22,35)(29,30,40);(10,12,13,20)(11,16,17,29)(14,21,23,34)(15,22,24,35)(18,30,27,38)(19,31,25,36)(26,37,28,39)(32,40,33,41);(7,9)(10,11)(12,16)(13,17)(14,25)(15,27)(18,24)(19,23)(20,29)(21,36)(22,38)(26,33)(28,32)(30,35)(31,34)(37,41)(39,40)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236806);