C₈₉×#7323 #36385171

order	22784
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₈₉×#7323

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#342
commutator	C₂²×C₄	C₄²×C₈₉
Frattini, cosocle	C₂×C₄×Q₈	C₂²×C₈₉
prime radical	C₂³×C₈₉	#95
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ C₂×C₄×Q₈×C₈₉ ◁ G

256 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 11392

A minimal faithful permutation representation — 2 generators on 113 points:

(1,2,3,6,4,5,7,8)(10,14,15,21)(16,24,23,22)(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);(2,5)(3,7)(9,10,12,16,13,15,20,23)(11,14,17,22,18,21,19,24)

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, 311);