C₈₉×#7348 #36385196

order	22784
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₈₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₉×#7348

Subgroups

	subgroup	G/H
center, socle	C₂×C₈₉	#1317
commutator	C₂²×C₄	C₂×C₈×C₈₉
Frattini, cosocle	#381	C₂²×C₈₉
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂³×C₈₉ ◁ C₂⁴×C₈₉ ◁ #381×C₈₉ ◁ G

200 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Conjugacy classes & character table

2492 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 336);