C₅×#85469 #14511207

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

solvable

Aliases

direct product

C₅×#85469

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

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

normal subgroups only

This group has 508 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,28,21,15,22,29,11,13,23,7,9,27)(8,24,34,12,30,36,16,32,35,17,18,37)(10,14,20,26)(19,33,31,25);(7,12)(8,16)(13,24)(14,25)(15,17)(18,30)(19,31)(26,33)(27,36)(28,32)(29,37)(34,35);(9,20)(10,22)(13,26)(14,28)(18,31)(19,30)(21,23)(24,33)(25,32)(27,29)(34,35)(36,37)

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, 236905);