C₅×#73039 #14510763

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

solvable

Aliases

direct product

C₅×#73039

Splittings

Fitting

C₅×#504 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₅×C₈	C₂×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂×C₅×C₈
Frattini	C₈	C₂×C₅×S₄
Fitting	C₅×#504	D₃
socle	C₂×C₅	C₈×S₄
prime radical	C₅×#985	C₄
cosocle	#131⋊C₃	C₂²×C₅
hypercenter	C₅×C₁₆	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₅×C₈ ◁ C₅×C₁₆

170 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 40 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)(6,31,46,14,49,61,18,66,26,13,19,58,24,36,63,51,27,39,8,52,50,15,64,69,12,16,44,29,22,60,42,56,28,7,45,59,9,54,65,33,10,38,21,34,48,30,40,68)(11,17,20,35,23,37,41,57,25,32,43,53,47,55,62,67);(7,13)(10,25)(11,27)(14,29)(15,30)(16,17)(19,43)(20,45)(22,47)(23,49)(26,28)(31,32)(33,51)(34,35)(36,37)(40,62)(41,64)(44,46)(48,50)(52,53)(54,55)(56,57)(63,65)(66,67)

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, 236461);