C₇₃×#12837 #32922241

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₃×#12837

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇₃
Frattini, cosocle	C₂²×C₈	C₂³×C₇₃
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂⁴×C₇₃	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₃ ◁ C₂³×C₄×C₇₃ ◁ C₂³×C₈×C₇₃ ◁ G

394 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 166 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:

(1,2)(3,4)(5,6,9,14,11,15,10,25)(7,16,19,31,21,32,20,35)(8,22,23,29,24,36,17,13)(12,26,27,34,28,33,30,18)(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);(5,7,17,33,10,20,24,34,11,21,23,26,9,19,8,18)(6,12,22,31,14,27,29,32,15,28,36,35,25,30,13,16);(2,4)

Conjugacy classes & character table

5548 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 5825);