C₇₃×#12772 #32922176

order	18688
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₃×#12772

Subgroups

	subgroup	G/H
center	C₄×C₇₃	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×#41	C₂³×C₇₃
socle	C₂×C₇₃	#1790
prime radical	C₇₃×#3529	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₂×#41×C₇₃ ◁ G

760 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 2336

A minimal faithful permutation representation — 3 generators on 105 points:

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

Conjugacy classes & character table

2482 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 5760);