C₇₃×#7117 #32916521

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₃×#7117

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#329
commutator	C₄²	C₂×C₈×C₇₃
Frattini, cosocle	C₂×#97	C₂²×C₇₃
prime radical	C₂³×C₇₃	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×#97×C₇₃ ◁ G

230 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,7,8,5)(4,6)(9,10,12,17,13,18,26,36,14,19,25,37,28,16,33,34)(11,20,21,38,22,15,29,30,23,31,39,35,27,32,24,40)(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);(1,3)(2,5)(4,8)(6,7)(9,11,14,23)(10,15,19,32)(12,24,25,29)(13,27,28,22)(16,31,18,20)(17,35,37,38)(21,26,39,33)(30,34,40,36)

Conjugacy classes & character table

2920 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 105);