C₇₃×#7224 #32916628

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇₃×#7224

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#340
commutator	C₂×C₈	C₄²×C₇₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₃
prime radical	C₂³×C₇₃	#103
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

184 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 121
transitive degree	μ_t(G) = 9344

A minimal faithful permutation representation — 2 generators on 121 points:

(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,21,26,23,29,40,28)(19,30,32,43,35,45,34,48)(20,36,38,25,39,27,22,41)(24,33,42,31,44,47,46,37)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,22,33,40,32,39,31,23,35,38,47,21,34,20,37)(18,24,27,43,26,46,41,30,29,44,36,48,28,42,25,45)

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, 212);