C₇₉×#13421 #34125110

order	20224
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#13421

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂³×C₇₉
prime radical	C₂²×D₄×C₇₉	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×D₄×C₇₉ ◁ G

496 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 99
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 99 points:

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

Conjugacy classes & character table

2449 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6409);