C₇₉×#7491 #34119180

order	20224
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#7491

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₄²×C₇₉
Frattini, cosocle	C₂×C₄⋊Q₈	C₂²×C₇₉
prime radical	C₂³×C₇₉	#99
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×C₄⋊Q₈×C₇₉ ◁ G

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

normal subgroups only

This group has 276 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) = 99
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 2 generators on 99 points:

(5,6,8,12,10,13,18,15)(7,14,11,19,16,17,20,9)(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)(6,9)(8,12)(10,16)(11,17)(13,19)(14,20)(15,18)

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, 479);