C₇₉×#7499 #34119188

order	20224
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₉×#7499

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#353
commutator	C₄²	C₄²×C₇₉
Frattini, cosocle	C₂×C₄⋊Q₈	C₂²×C₇₉
prime radical	C₂³×C₇₉	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×C₄⋊Q₈×C₇₉ ◁ G

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

normal subgroups only

This group has 272 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) = 5056

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

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

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, 487);