D₁₉×#11725078 #45691547

order	33516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂×C₇

Aliases

direct product

D₁₉×#11725078

Splittings

derived, Hall	C₇²×C₁₉ ⋊ C₂²×C₉
Hall	C₁₉×#89205 ⋊ C₂²
Hall	D₁₉×C₇⋊D₇ ⋊ C₉
Hall	#89205 ⋊ C₂×D₁₉
Sylow	C₇² ⋊ C₂×C₉×D₁₉
Sylow	C₁₉ ⋊ C₂×#11725078

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₁₉×#64425
commutator, hypocenter	C₇²×C₁₉	C₂²×C₉
Fitting, socle, cosocle	C₃×C₇²×C₁₉	C₂²×C₃
prime radical	C₃×D₁₉×C₇⋊D₇	C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁ ◁ C₃

160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 42
transitive degree	μ_t(G) = 1197

A minimal faithful permutation representation — 2 generators on 42 points:

(2,5,6,3,4,7)(9,13,12,10,11,14)(15,19,23,17,22,18,20,16,21)(24,25,27,29,31,33,35,37,39,41,42,40,38,36,34,32,30,28,26);(1,2,4,7,6,5,3)(8,9,12,13,11,14,10)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33516, 345);