C₅×#85475 #14511213

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product

C₅×#85475

Splittings

Fitting

C₅×#578 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	D₄×S₄
commutator, cosocle	#41⋊C₃	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#578	D₃
hypercenter	C₅×D₈	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×D₈

490 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 3 generators on 69 points:

(1,2,3,4,5)(6,29,31)(7,16,41,14,30,40)(8,46,50,19,49,67)(9,25,57,24,52,55)(10,28,12)(13,62,61,33,42,69)(15,56,65)(17,27)(18,58,60,37,34,68)(20,21,66,23,44,48)(22,45)(26,53)(35,59)(36,38,64)(43,63)(47,51);(6,8)(7,13)(9,19)(10,21)(11,22)(12,23)(14,18)(15,37)(16,34)(17,35)(20,24)(25,44)(26,45)(27,43)(28,48)(29,49)(30,42)(31,50)(32,51)(33,36)(38,62)(39,63)(40,60)(41,61)(46,52)(47,53)(54,59)(55,66)(56,58)(57,67)(64,69)(65,68);(6,7)(8,18)(9,15)(10,27)(11,30)(12,14)(13,23)(16,32)(17,29)(19,33)(20,37)(21,35)(22,34)(24,36)(25,54)(26,56)(28,41)(31,40)(38,53)(39,52)(42,51)(43,49)(44,63)(45,62)(46,59)(47,58)(48,60)(50,61)(55,65)(57,64)(66,69)(67,68)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236911);