C₈₉×#7225 #36385073

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₈₉×#7225

Subgroups

	subgroup	G/H
center	C₄×C₈₉	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₈₉
Frattini, cosocle	C₄×M₁₆	C₂²×C₈₉
socle	C₂×C₈₉	#1288
prime radical	C₂³×C₈₉	#104
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂×C₈×C₈₉ ◁ G

192 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 105
transitive degree	μ_t(G) = 1424

A minimal faithful permutation representation — 2 generators on 105 points:

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

Conjugacy classes & character table

3560 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 213);