C₇₉×#12531 #34124220

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12525
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₉×#12531

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₉
prime radical	#493×C₇₉	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ C₈²×C₇₉ ◁ G

272 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 10112

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

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

Conjugacy classes & character table

3634 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5519);