C₅×#73139 #14510863

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

solvable

Aliases

direct product

C₅×#73139

Splittings

Fitting

C₅×#578 ⋊ D₃

Subgroups

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

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

normal subgroups only

This group has 190 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)(7,51,42,13,30,61)(8,64,52,19,50,69)(9,25,57,20,46,56)(10,28,12)(14,54,65,15,59,67)(16,60,18,34,41,36)(17,35)(21,68,23,43,49,45)(22,44)(24,37,66,33,39,63)(26,47)(27,48)(38,40)(53,62)(55,58);(6,7)(8,13)(9,24)(10,27)(11,30)(12,18)(14,19)(15,20)(16,32)(17,29)(21,48)(22,51)(23,36)(25,40)(26,39)(28,42)(31,41)(33,45)(34,53)(35,50)(37,62)(38,64)(43,58)(44,59)(46,55)(47,54)(49,61)(52,60)(56,66)(57,67)(63,69)(65,68)

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, 236561);