C₅×#73040 #14510764

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₅×#73040

Splittings

Fitting

C₅×#504 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#5402
commutator, cosocle	#131⋊C₃	C₂²×C₅
Frattini	C₈	C₂×C₅×S₄
Fitting	C₅×#504	D₃
hypercenter	C₅×C₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₅×D₈
LCS	G ▷ #131⋊C₃ ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈ ◁ C₅×C₁₆

176 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 32 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)(6,42,63,30,46,67,22,31,26,18,34,68,19,38,52,13,27,44,14,64,55,15,20,60,12,16,62,37,23,66,8,43,28,7,51,69,9,56,49,33,10,41,36,58,54,29,50,61)(11,17,35,59,24,39,53,40,25,45,48,65,47,57,21,32);(7,13)(8,14)(9,19)(10,25)(11,27)(15,37)(16,40)(17,43)(18,33)(20,48)(21,51)(22,36)(23,24)(26,28)(29,30)(31,45)(32,42)(34,53)(35,50)(38,59)(39,58)(41,61)(44,60)(46,47)(49,63)(52,62)(56,65)(57,64)(66,68)(67,69)

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, 236462);