C₈₃×#8474 #35236298

order	21248
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₈₃×#8474

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₃	#379
commutator	C₂²×C₄	C₂²×C₄×C₈₃
Frattini, cosocle	C₂×C₄²	C₂³×C₈₃
prime radical	C₂×#41×C₈₃	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂×C₄²×C₈₃ ◁ G

682 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 10624

A minimal faithful permutation representation — 3 generators on 115 points:

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

Conjugacy classes & character table

3320 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 1462);