C₈₃×#7409 #35235233

order	21248
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₃×#7409

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₃	AΣL(1,𝔽₁₆)
commutator	C₂×C₄⋊C₄	C₂×C₄×C₈₃
Frattini, cosocle	#391	C₂²×C₈₃
prime radical	#537×C₈₃	C₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂³×C₈₃ ◁ C₂²×C₄×C₈₃ ◁ #391×C₈₃ ◁ G

268 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 2656

A minimal faithful permutation representation — 2 generators on 107 points:

(1,2)(3,5)(4,7)(6,8)(9,10,12,16,14,11,18,20)(13,21,23,19,24,17,22,15)(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);(2,5)(3,8)(6,7)(9,11)(10,14)(12,19)(13,17)(15,18)(16,22)(20,23)(21,24)

Conjugacy classes & character table

2573 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 397);