C₈₃×#12884 #35240708

order	21248
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₈₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₈₃×#12884

Subgroups

	subgroup	G/H
center	C₂×C₈×C₈₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₈₃
Frattini, cosocle	C₂²×C₈	C₂³×C₈₃
socle	C₂²×C₈₃	C₂×C₄⋊C₈
prime radical	C₂³×C₈₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₈₃ ◁ C₂×C₄×C₈×C₈₃ ◁ G

302 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 123
transitive degree	μ_t(G) = 5312

A minimal faithful permutation representation — 3 generators on 123 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,20,15,19,31,36,16,21,30,18,13,32,12,28)(11,22,25,37,26,34,23,35,27,38,39,40,24,17,29,33)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(2,5)(6,8)(9,11,13,24,16,27,15,26)(10,17,19,22,21,34,32,38)(12,29,30,39,31,23,14,25)(18,35,28,40,20,33,36,37);(1,3,4,7)(2,6,5,8)(9,12,13,30,16,31,15,14)(10,18,19,28,21,20,32,36)(11,23,24,25,27,29,26,39)(17,33,22,37,34,35,38,40)

Conjugacy classes & character table

5312 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 5872);