C₈₃×#11579 #35239403

order	21248
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₈₃×#11579

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₃	C₂×#99
commutator	C₂⁴	C₂²×C₄×C₈₃
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₈₃
prime radical	C₂²×D₄×C₈₃	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂²×C₄×C₈₃ ◁ C₂×C₄×D₄×C₈₃ ◁ G

424 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 5312

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

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

Conjugacy classes & character table

3071 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21248, 4567);