C₅×#73130 #14510854

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#73130

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	(C₂×S₄)⋊C₄
commutator	C₂×SL(2,𝔽₃)	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₅×#568	D₃
prime radical	C₂×C₅×#41⋊C₃	C₂²
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₅×M₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₂²⋊C₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×M₁₆

302 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 48 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)(6,46,54,19,10,48,26,44,31,8,53,67,12,21,52,43,29,49,9,64,28,20,24,66)(7,41,61,33,15,58,23,56,39,13,50,69,17,34,60,40,30,59,14,62,38,18,36,68)(11,22,25,45,32,47,55,65)(16,35,37,57,27,42,51,63);(6,7,12,17)(8,18,20,13)(9,23,26,14)(10,27,29,16)(11,30,32,15)(19,33,43,40)(21,35,46,42)(22,34,47,41)(24,37,53,51)(25,36,55,50)(28,39,31,38)(44,63,64,57)(45,62,65,56)(48,58,49,59)(52,60,54,61)(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, 236552);