C₈₃×#7341 #35235165

order	21248
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₈₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₃×#7341

Subgroups

	subgroup	G/H
center, socle	C₂×C₈₃	#1316
commutator	C₂⁴	C₂×C₈×C₈₃
Frattini, cosocle	#407	C₂²×C₈₃
prime radical	C₂⁴×C₈₃	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₃ ◁ C₂×C₄×C₈₃ ◁ C₂²×C₄×C₈₃ ◁ #407×C₈₃ ◁ G

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

normal subgroups only

This group has 324 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) = 115
transitive degree	μ_t(G) = 2656

A minimal faithful permutation representation — 2 generators on 115 points:

(1,2,4,9,5,10,17,24,6,8,16,22,19,23,29,31)(3,11,12,26,13,25,27,32,14,7,15,20,18,21,28,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,3,6,14)(2,7,8,11)(4,15,16,12)(5,18,19,13)(9,20,22,26)(10,25,23,21)(17,27,29,28)(24,32,31,30)

Conjugacy classes & character table

2324 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 329);