C₇₉×#13458 #34125147

order	20224
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#13458

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
prime radical	C₂³×C₇₉	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

324 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 115
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 115 points:

(5,6,8,11)(7,15)(9,21,20,35)(10,24,17,14)(12,28)(13,31,27,23)(16,33)(18,26)(19,34)(22,30)(25,32)(29,36)(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);(1,2)(3,4)(6,11)(7,16,19,25)(12,29,26,30)(13,27)(14,24)(15,33,34,32)(18,22,28,36)(21,35);(2,4)(5,7,17,16,8,19,10,25)(6,12,24,30,11,26,14,29)(9,22,31,28,20,36,23,18)(13,32,35,34,27,33,21,15)

Conjugacy classes & character table

3160 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6446);