C₇₉×#12311 #34124000

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12310
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#12311

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₉
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂×C₄×C₇₉ ◁ C₄×C₈×C₇₉ ◁ G

366 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Conjugacy classes & character table

2686 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5299);