C₇₃×#8225 #32917629

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₃×#8225

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₄³×C₇₃ ◁ G

444 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 18688

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

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

Conjugacy classes & character table

3796 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 1213);