C₅×#73140 #14510864

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

solvable

Aliases

direct product

C₅×#73140

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#5415
commutator, cosocle	#131⋊C₃	C₂²×C₅
Frattini	C₈	C₂×C₅×S₄
Fitting	C₅×#578	D₃
prime radical	C₅×#578⋊C₃	C₂
hypercenter	C₅×D₈	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×D₈

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

normal subgroups only

This group has 184 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)(6,29,31)(7,51,43,13,30,61)(8,62,52,19,50,69)(9,25,57,20,46,56)(10,28,12)(14,54,65,15,59,67)(16,60,18,34,42,36)(17,35)(21,68,23,44,49,37)(22,45)(24,38,66,33,40,64)(26,47)(27,48)(39,41)(53,63)(55,58);(6,7,12,18)(8,13,23,36)(9,24,20,15)(10,27,29,17)(11,30,32,16)(14,37,33,19)(21,48,50,35)(22,51,53,34)(25,41,46,55)(26,40,47,54)(28,43,31,42)(38,45,59,63)(39,44,58,62)(49,61,52,60)(56,66,57,67)(64,68,65,69)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236562);