C₈₃×#12344 #35240168

order	21248
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₃×#12344

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₃	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₈₃
prime radical	C₈₃×#3334	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₄²×C₈₃ ◁ C₈×D₄×C₈₃ ◁ G

330 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 107
transitive degree	μ_t(G) = 10624

A minimal faithful permutation representation — 3 generators on 107 points:

(2,3)(4,5)(7,8)(9,10,13,17)(11,14,20,23)(12,21,19,16)(15,24,22,18)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(1,2,6,8)(3,4,7,5)(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22);(1,3)(2,4)(5,8)(6,7)

Conjugacy classes & character table

3320 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 5332);