C₇₃×#7350 #32916754

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₃×#7350

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₃	#1317
commutator	C₂²×C₄	C₂×C₈×C₇₃
Frattini, cosocle	#407	C₂²×C₇₃
prime radical	C₂⁴×C₇₃	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #407×C₇₃ ◁ G

178 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 105
transitive degree	μ_t(G) = 2336

A minimal faithful permutation representation — 2 generators on 105 points:

(1,2,5,10,6,9,19,25,7,11,18,28,4,16,17,27)(3,12,14,29,13,8,21,22,15,24,30,32,20,23,31,26)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,3)(2,8,11,23)(4,13)(5,14)(6,20)(7,15)(9,12,16,24)(10,26,28,22)(17,21)(18,30)(19,31)(25,32,27,29)

Conjugacy classes & character table

2044 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 338);