C₈₃×#7518 #35235342

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

Aliases

direct product

C₈₃×#7518

Subgroups

	subgroup	G/H
center, socle	C₂×C₈₃	#1402
commutator	#127	C₂×C₄×C₈₃
Frattini, cosocle	#492	C₂²×C₈₃
prime radical	#127×C₈₃	C₂×C₄
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₃ ◁ C₂²×C₈₃ ◁ C₂×C₄×C₈₃ ◁ C₄²×C₈₃ ◁ #492×C₈₃ ◁ G

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

normal subgroups only

This group has 148 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) = 115
transitive degree	μ_t(G) = 2656

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

(1,2,4,9,6,11,18,27)(3,12,13,25,15,22,29,32)(5,8,16,24,19,10,17,26)(7,20,21,30,23,28,31,14)(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);(1,3,6,15)(2,7,11,23)(4,13,18,29)(5,14,19,30)(8,12,10,22)(9,25,27,32)(16,20,17,28)(21,24,31,26)

Conjugacy classes & character table

1826 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 506);