C₈₉×#11540 #36389388

order	22784
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₉×#11540

Subgroups

	subgroup	G/H
center	C₄×C₈₉	#410
commutator	C₂²×C₄	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×M₁₆	C₂³×C₈₉
socle	C₂×C₈₉	#1467
prime radical	C₂³×C₈₉	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂×C₈×C₈₉ ◁ #405×C₈₉ ◁ G

262 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 2848

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

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

Conjugacy classes & character table

3026 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 4528);