C₅×#73120 #14510844

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₅×#73120

Subgroups

	subgroup	G/H
center	C₄×C₅	C₄⋊S₄
commutator	#41⋊C₃	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₂×C₅×#131	D₃
socle	C₂×C₅	C₄⋊SL(2,ℤ/4)
prime radical	C₂×C₅×#41⋊C₃	C₂²
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₂×C₅×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₄⋊C₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

236 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 69 points:

(1,2,3,4,5)(6,52,57,21,11,51,9,48,34,8,25,68,13,22,59,20,32,54,10,46,31,24,28,69)(7,61,65,19,16,62,27,43,42,14,58,67,18,36,64,45,33,63,15,53,41,38,39,66)(12,23,29,49,35,55,26,47)(17,37,40,44,30,60,56,50);(6,7,9,15,13,18,10,27)(8,19,20,38,24,45,21,14)(11,30,25,56,32,17,28,40)(12,33,26,58,35,16,29,39)(22,50,46,37,52,44,48,60)(23,53,47,36,55,43,49,61)(31,42,57,65,34,41,59,64)(51,67,68,62,54,66,69,63)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236542);