C₇₃×#7366 #32916770

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#7366

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#354
commutator	C₄²	C₂×C₈×C₇₃
Frattini, cosocle	#382	C₂²×C₇₃
prime radical	C₂³×C₇₃	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂³×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #382×C₇₃ ◁ G

158 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 113
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 2 generators on 113 points:

(1,2,3,5,4,6,7,8)(9,10)(11,19,21,32)(12,23,24,39)(13,26,27,17)(14,18)(15,20,28,35)(16,25)(22,34,38,33)(29,36,30,37)(31,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(9,11,13,21,14,22,27,38)(10,15,17,29,18,30,26,28)(12,20,25,36,24,37,40,35)(16,19,23,33,31,34,39,32)

Conjugacy classes & character table

2482 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 354);