C₈₉×#7427 #36385275

order	22784
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₈₉×#7427

Subgroups

	subgroup	G/H
center, socle	C₂×C₈₉	#1351
commutator	C₂×C₂²⋊C₄	C₂×C₄×C₈₉
Frattini, cosocle	#381	C₂²×C₈₉
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂³×C₈₉ ◁ C₂⁴×C₈₉ ◁ #381×C₈₉ ◁ G

214 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 121
transitive degree	μ_t(G) = 2848

A minimal faithful permutation representation — 2 generators on 121 points:

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

Conjugacy classes & character table

2225 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22784, 415);