C₅×#69058 #14318598

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#69058

Splittings

Fitting	C₃×C₄×C₅×C₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #2240
Sylow	C₃ ⋊ C₅×#2240

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×C₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₅×#4695	C₂
hypercenter	C₄×C₅×C₁₆	D₃
hypocenter	C₃	C₅×#2240
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆

320 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 32
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 44296);