C₈₉×#11496 #36389344

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

Aliases

direct product

C₈₉×#11496

Subgroups

	subgroup	G/H
center	C₂×C₈×C₈₉	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₈₉
Frattini, cosocle	C₂²×C₈	C₂³×C₈₉
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₈₉	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

296 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 11392

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

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,24,27,36,44,23,29,35,45,21,37,20,26)(19,30,33,40,32,25,39,41,34,43,47,48,38,42,31,46)(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);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,34)(18,25,29,42)(20,31,36,39)(21,38,24,32)(22,33,35,47)(26,40,44,48)(27,43,37,30)(28,41,45,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,35,23,36,24,22)(18,26,37,45,29,44,27,28)(19,31,38,47,34,39,32,33)(25,40,30,46,42,48,43,41)

Conjugacy classes & character table

6764 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 4484);