C₇₃×#12679 #32922083

order	18688
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₇₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₇₃×#12679

Subgroups

	subgroup	G/H
center	C₂²×C₈×C₇₃	C₂³
commutator	C₂²	C₂²×C₁₆×C₇₃
Frattini, cosocle	C₂²×C₈	C₂³×C₇₃
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂⁵×C₇₃	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₇₃ ◁ G

1198 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 1198 conjugacy classes of subgroups — too many to draw. Only its 286 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:

(1,2)(3,4)(5,6)(7,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24)(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);(2,4);(6,8)

Conjugacy classes & character table

8176 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 5667);