C₈₉×#9508 #36387356

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₈₉×#9508

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#386
commutator	C₂⁴	C₂²×C₄×C₈₉
Frattini, cosocle	C₂³×C₄	C₂³×C₈₉
prime radical	C₂²×D₄×C₈₉	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ G

542 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 113
transitive degree	μ_t(G) = 5696

A minimal faithful permutation representation — 3 generators on 113 points:

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

Conjugacy classes & character table

3293 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 2496);