C₈₃×#7334 #35235158

order	21248
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂×C₈₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₃×#7334

Subgroups

	subgroup	G/H
center	C₂×C₈×C₈₃	C₂²⋊C₄
commutator	C₂²	C₂×C₃₂×C₈₃
Frattini, cosocle	C₂²×C₁₆	C₂²×C₈₃
socle	C₂²×C₈₃	C₂²⋊C₁₆
prime radical	C₂⁴×C₈₃	C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₈₃ ◁ C₂²×C₁₆×C₈₃ ◁ G

182 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 123 points:

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

Conjugacy classes & character table

7304 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 322);