C₇₃×#21828 #32931232

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₇₃×#21828

Subgroups

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

990 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 366 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) = 105
transitive degree	μ_t(G) = 4672

A minimal faithful permutation representation — 4 generators on 105 points:

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

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, 14816);