C₈₉×#7528 #36385376

order	22784
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₈₉×#7528

Subgroups

	subgroup	G/H
center, socle	C₂×C₈₉	#1408
commutator	C₄×C₈	C₂×C₄×C₈₉
Frattini, cosocle	#492	C₂²×C₈₉
prime radical	#127×C₈₉	D₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂²×C₈₉ ◁ C₂×C₄×C₈₉ ◁ C₄²×C₈₉ ◁ #492×C₈₉ ◁ G

142 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 30 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 — 2 generators on 121 points:

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

Conjugacy classes & character table

1691 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 516);