C₈₉×#9757 #36387605

order	22784
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈₉ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₈₉×#9757

Subgroups

	subgroup	G/H
center, socle	C₂²×C₈₉	#390
commutator	C₄²	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	#532×C₈₉	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

400 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 137
transitive degree	μ_t(G) = 5696

A minimal faithful permutation representation — 3 generators on 137 points:

(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18)(19,23,28,37)(20,30)(21,34)(22,25)(24,32)(26,31)(27,39,41,48)(29,43,40,36)(33,44)(35,46,47,42)(38,45)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19)(18,23,25,37)(20,27)(21,29)(22,28)(24,35,38,47)(26,36,34,43)(30,39,44,48)(31,40)(32,42)(33,41)(45,46);(2,7)(6,14)(8,12)(9,15)(17,20,31,45,22,33,21,32)(18,24,34,44,25,38,26,30)(19,27,40,46,28,41,29,42)(23,35,43,48,37,47,36,39)

Conjugacy classes & character table

4628 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 2745);