C₈₃×#7380 #35235204

order	21248
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₈₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₈₃×#7380

Subgroups

	subgroup	G/H
center	C₂×C₈×C₈₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₁₆×C₈₃
Frattini, cosocle	C₄×C₁₆	C₂²×C₈₃
socle	C₂²×C₈₃	C₂²⋊C₁₆
prime radical	C₂×D₄×C₈₃	C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₈₃ ◁ C₄×C₁₆×C₈₃ ◁ G

128 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 10624

A minimal faithful permutation representation — 2 generators on 123 points:

(1,2)(3,5)(4,7)(6,8)(9,10,11,16,13,17,20,28,14,18,21,29,24,30,33,38,12,15,19,25,22,26,31,35,23,27,32,36,34,37,39,40)(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);(2,5)(3,8)(6,7)(10,15)(11,19)(17,26)(18,27)(20,31)(21,32)(30,37)(33,39)

Conjugacy classes & character table

6308 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 368);