C₅×#85471 #14511209

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

solvable

Aliases

direct product

C₅×#85471

Splittings

Fitting

C₅×#577 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	D₄×S₄
commutator, cosocle	#41⋊C₃	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#577	D₃
hypercenter	C₅×SD₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×SD₁₆

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

normal subgroups only

This group has 488 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)(6,28,30)(7,16,41,14,29,40)(8,44,50,18,48,67)(9,24,57,23,52,55)(10,27,12)(13,62,61,32,49,69)(15,56,65)(17,26)(19,20,66,22,42,47)(21,43)(25,53)(33,68,35,58,60,37)(34,59)(36,38,64)(45,51)(46,63);(6,8,12,22)(7,13,14,35)(9,18,23,19)(10,20,28,48)(11,21,31,51)(15,37,36,32)(16,33,29,49)(17,34,26,46)(24,42,52,44)(25,43,53,45)(27,47,30,50)(38,62,56,58)(39,63,54,59)(40,60,41,61)(55,66,57,67)(64,69,65,68);(6,7)(8,13)(9,15)(10,26)(11,29)(12,14)(16,31)(17,28)(18,37)(19,32)(20,46)(21,49)(22,35)(23,36)(24,54)(25,56)(27,41)(30,40)(33,51)(34,48)(38,53)(39,52)(42,59)(43,58)(44,63)(45,62)(47,61)(50,60)(55,65)(57,64)(66,68)(67,69)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236907);