C₇₉×#12304 #34123993

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#12304

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₇₉
prime radical	C₇₉×#3259	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₂×M₁₆×C₇₉ ◁ G

286 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 111
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 111 points:

(1,2,7,11,6,12,5,22)(3,13,23,30,15,9,16,26)(4,17,18,27,20,8,21,10)(14,24,31,25,19,28,32,29)(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);(2,8)(4,18)(5,7)(9,24)(10,22)(11,27)(12,17)(13,28)(14,31)(16,23)(19,32)(20,21)(25,30)(26,29);(1,3,6,15)(2,9)(4,19,20,14)(5,16,7,23)(8,24)(10,25)(11,26)(12,13)(17,28)(18,32,21,31)(22,30)(27,29)

Conjugacy classes & character table

1975 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5292);