C₈₃×#7351 #35235175

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

Aliases

direct product

C₈₃×#7351

Subgroups

	subgroup	G/H
center, socle	C₂×C₈₃	#1317
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

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

normal subgroups only

This group has 178 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:

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

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, 339);