C₈₉×#12054 #36389902

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₈₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₈₉×#12054

Subgroups

	subgroup	G/H
center	C₂×C₈×C₈₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₈₉
Frattini, cosocle	C₄×C₈	C₂³×C₈₉
socle, prime radical	C₂²×C₈₉	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₈₉ ◁ C₄×C₁₆×C₈₉ ◁ G

206 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 121
transitive degree	μ_t(G) = 22784

A minimal faithful permutation representation — 3 generators on 121 points:

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

Conjugacy classes & character table

7832 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 5042);