C₇₃×#7270 #32916674

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#7270

Subgroups

	subgroup	G/H
center	C₄×C₇₃	#331
commutator	C₄²	C₄²×C₇₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₃
socle	C₂×C₇₃	#1296
prime radical	C₂³×C₇₃	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

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

normal subgroups only

This group has 208 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) = 105
transitive degree	μ_t(G) = 2336

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

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

Conjugacy classes & character table

3358 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 258);