C₅×#73123 #14510847

order	1920
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₄

solvable

Aliases

direct product

C₅×#73123

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₄²×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₂×C₅×#131	D₃
socle	C₂²×C₅	C₄×S₄
prime radical	C₂×C₅×#41⋊C₃	C₂²
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₂×C₅×C₈	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

286 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 41
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5)(10,28,37,17,13,32,22,30,26,11,36,41,15,18,35,27,25,33,12,38,24,16,20,40)(14,19,21,31,23,29,34,39);(6,7,8,9)(11,16)(13,23)(14,25)(17,27)(18,19)(20,34)(21,36)(24,26)(28,29)(30,31)(35,37)(38,39)

Conjugacy classes & character table

260 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236545);