C₅×#73119 #14510843

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₅×#73119

Subgroups

	subgroup	G/H
center	C₅×C₈	C₂×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂×C₅×C₈
Frattini	C₈	C₂×C₅×S₄
Fitting	C₂×C₅×#131	D₃
socle	C₂×C₅	C₂×A₄⋊C₈
prime radical	C₂×C₅×#41⋊C₃	C₄
cosocle	#131⋊C₃	C₂²×C₅
hypercenter	C₂×C₅×C₈	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×C₈

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

normal subgroups only

This group has 192 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)(6,27,29)(7,16,41,13,28,39)(8,40,48)(9,51,53)(10,26,12)(14,34,59,31,47,58)(15,36,63,32,52,61)(17,25)(18,62,67)(19,46,21)(22,50,24)(33,56,69,55,66,68)(35,45)(37,49)(42,65,44)(57,64);(6,7,8,14,9,15,18,33,12,13,21,31,24,32,44,55)(10,25,19,45,22,49,42,64,27,17,40,35,51,37,62,57)(11,28,20,47,23,52,43,66,30,16,38,34,54,36,60,56)(26,41,46,59,50,63,65,69,29,39,48,58,53,61,67,68)

Conjugacy classes & character table

260 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236541);