C₈₃×#7529 #35235353

order	21248
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈₃ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₈₃×#7529

Subgroups

	subgroup	G/H
center, socle	C₂×C₈₃	#1408
commutator	C₄×C₈	C₂×C₄×C₈₃
Frattini, cosocle	#493	C₂²×C₈₃
prime radical	C₈₃×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₃ ◁ C₂²×C₈₃ ◁ C₂×C₄×C₈₃ ◁ C₄²×C₈₃ ◁ #493×C₈₃ ◁ G

186 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 99
transitive degree	μ_t(G) = 1328

A minimal faithful permutation representation — 2 generators on 99 points:

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

Conjugacy classes & character table

2075 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 517);