C₅×#69112 #14316067

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

Aliases

direct product

C₅×#69112

Splittings

Fitting	C₃×C₅×C₂²⋊C₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #2185
Sylow	C₃ ⋊ C₅×#2185

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×C₂²⋊C₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₅×#5115	C₂
hypercenter	C₅×C₂²⋊C₁₆	D₃
hypocenter	C₃	C₅×#2185
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₅×C₂²⋊C₁₆

452 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 28
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 28 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 41765);