C₇₃×#7374 #32916778

order	18688
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇₃×#7374

Subgroups

	subgroup	G/H
center	C₄×C₇₃	#353
commutator	C₄²	C₂×C₈×C₇₃
Frattini, cosocle	#423	C₂²×C₇₃
socle	C₂×C₇₃	#1325
prime radical	C₂³×C₇₃	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ #423×C₇₃ ◁ G

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

normal subgroups only

This group has 148 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:

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

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, 362);