C₇₃×#13258 #32922662

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₃×#13258

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇₃
Frattini, cosocle	C₂²×C₈	C₂³×C₇₃
prime radical	C₂²×D₈×C₇₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

482 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 109
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 3 generators on 109 points:

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

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, 6246);