C₈₉×#7259 #36385107

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

Aliases

direct product

C₈₉×#7259

Subgroups

	subgroup	G/H
center, socle	C₂³×C₈₉	#102
commutator	C₂×C₈	C₄²×C₈₉
Frattini, cosocle	C₂³×C₈	C₂²×C₈₉
prime radical	C₂⁴×C₈₉	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ C₂³×C₈×C₈₉ ◁ G

342 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 117
transitive degree	μ_t(G) = 11392

A minimal faithful permutation representation — 2 generators on 117 points:

(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(14,19)(15,23)(16,17)(20,27)(21,24)(22,28)(25,26)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(6,9,11,10)(13,14,16,20,18,22,17,21)(15,24,23,28,26,27,25,19)

Conjugacy classes & character table

5162 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 247);