C₇₉×#7489 #34119178

order	20224
nilpotency class	5
isoclinism	isoclinic class of #1404
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#7489

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₄²×C₇₉
Frattini, cosocle	C₂×#124	C₂²×C₇₉
prime radical	C₂²×D₄×C₇₉	D₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×#124×C₇₉ ◁ G

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

normal subgroups only

This group has 302 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:

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

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, 477);