C₁₉×#942 #14269696

order	1824
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#942

Splittings

Fitting	C₂×C₃×Q₈×C₁₉ ⋊ C₂
Hall	C₃×C₁₉ ⋊ #137
Sylow	C₃ ⋊ C₁₉×#137

Subgroups

	subgroup	G/H
center	C₂×C₁₉	C₂×#67
commutator, cosocle	C₃×C₄	C₂³×C₁₉
Frattini	C₄	C₂²×D₃×C₁₉
Fitting	C₂×C₃×Q₈×C₁₉	C₂
socle	C₂×C₃×C₁₉	C₂×D₄
prime radical	C₁₉×#235	C₂
hypercenter	C₂×Q₈×C₁₉	D₃
hypocenter	C₃	C₁₉×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×Q₈×C₁₉

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 38
transitive degree	μ_t(G) = 912

A minimal faithful permutation representation — 3 generators on 38 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1824, 923);