C₈₉×#7521 #36385369

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

Aliases

direct product

C₈₉×#7521

Subgroups

	subgroup	G/H
center, socle	C₂×C₈₉	#1402
commutator	C₄⋊Q₈	C₂×C₄×C₈₉
Frattini, cosocle	#494	C₂²×C₈₉
prime radical	#555×C₈₉	C₄
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂²×C₈₉ ◁ C₂×C₄×C₈₉ ◁ C₄²×C₈₉ ◁ #494×C₈₉ ◁ G

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

normal subgroups only

This group has 148 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:

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

Conjugacy classes & character table

1958 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 509);