C₇₃×#9391 #32918795

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#9391

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₃	#386
commutator	C₂²×C₄	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂²×D₄×C₇₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

538 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 97
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 3 generators on 97 points:

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

Conjugacy classes & character table

3577 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 2379);